# Global stability of first-order methods for coercive tame functions

arXiv: https://arxiv.org/abs/2308.00899 · HTML: https://mlsanity.com/paper/2308.00899/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Aug 2023.

## Built on by (most cited first, 6 of 6)

- [SAD Neural Networks: Divergent Gradient Flows and Asymptotic Optimality via o-minimal Structures](https://mlsanity.com/paper/2505.09572/) (2025)
- [On exploration of an interior mirror descent flow for stochastic nonconvex constrained problem](https://mlsanity.com/paper/2507.15264/) (2025)
- [Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2404.09438/) (2024)
- [Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions](https://mlsanity.com/paper/2403.11565/) (2024)
- [Inexact subgradient methods for semialgebraic functions](https://mlsanity.com/paper/2404.19517/) (2024)
- [The late-stage training dynamics of (stochastic) subgradient descent on homogeneous neural networks](https://mlsanity.com/paper/2502.05668/) (2025)

## Abstract (arXiv)

We consider first-order methods with constant step size for minimizing locally Lipschitz coercive functions that are tame in an o-minimal structure on the real field. We prove that if the method is approximated by subgradient trajectories, then the iterates eventually remain in a neighborhood of a connected component of the set of critical points. Under suitable method-dependent regularity assumptions, this result applies to the subgradient method with momentum, the stochastic subgradient method with random reshuffling and momentum, and the random-permutations cyclic coordinate descent method.