# Lipschitz minimization and the Goldstein modulus

arXiv: https://arxiv.org/abs/2405.12655 · HTML: https://mlsanity.com/paper/2405.12655/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 May 2024.

## Built on by (most cited first, 1 of 1)

- [Algorithmic Optimality Guarantees for Nonsmooth H_∞ Output-Feedback Policy Search](https://mlsanity.com/paper/2609.06197/) (2026)

## Abstract (arXiv)

Goldstein's 1977 idealized iteration for minimizing a Lipschitz objective fixes a distance - the step size - and relies on a certain approximate subgradient. That "Goldstein subgradient" is the shortest convex combination of objective gradients at points within that distance of the current iterate. A recent implementable Goldstein-style algorithm allows a remarkable complexity analysis (Zhang et al. 2020), and a more sophisticated variant (Davis and Jiang, 2022) leverages typical objective geometry to force near-linear convergence. To explore such methods, we introduce a new modulus, based on Goldstein subgradients, that robustly measures the slope of a Lipschitz function. We relate near-linear convergence of Goldstein-style methods to linear growth of this modulus at minimizers. We illustrate the idea computationally with a simple heuristic for Lipschitz minimization.