# Gradient sampling algorithm for subsmooth functions

arXiv: https://arxiv.org/abs/2503.16638 · HTML: https://mlsanity.com/paper/2503.16638/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Mar 2025.

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

- [The Normalized Maximum Likelihood for Regular Non-Smooth Models: Measure-Theoretic Foundations and Geometric Sampling](https://mlsanity.com/paper/2605.24477/) (2026)

## Abstract (arXiv)

This paper considers non-smooth optimization problems where we seek to minimize the pointwise maximum of a continuously parameterized family of functions. Since the objective function is given as the solution to a maximization problem, neither its values nor its gradients are available in closed form, which calls for approximation. Our approach hinges upon extending the so-called gradient sampling algorithm, which approximates the Clarke generalized gradient of the objective function at a point by sampling its derivative at nearby locations. This allows us to select descent directions around points where the function may fail to be differentiable and establish algorithm convergence to a stationary point from any initial condition. Our key contribution is to prove this convergence by alleviating the requirement on continuous differentiability of the objective function on an open set of full measure. We further provide assumptions under which a desired convex subset of the decision space is rendered attractive for the iterates of the algorithm.