# Subgradient sampling for nonsmooth nonconvex minimization

arXiv: https://arxiv.org/abs/2202.13744 · HTML: https://mlsanity.com/paper/2202.13744/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2022.

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

- [Adam-family Methods with Decoupled Weight Decay in Deep Learning](https://mlsanity.com/paper/2310.08858/) (2023)
- [A Functional Model Method for Nonconvex Nonsmooth Conditional Stochastic Optimization](https://mlsanity.com/paper/2405.10815/) (2024)
- [Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2404.09438/) (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)
- [Lipschitzian SLLNs for random functions](https://mlsanity.com/paper/2607.20411/) (2026)

## Abstract (arXiv)

Risk minimization for nonsmooth nonconvex problems naturally leads to first-order sampling or, by an abuse of terminology, to stochastic subgradient descent. We establish the convergence of this method in the path-differentiable case and describe more precise results under additional geometric assumptions. We recover and improve results from Ermoliev and Norkin [Cybern. Syst. Anal., 34 (1998), pp. 196--215] by using a different approach: conservative calculus and the ODE method. In the definable case, we show that first-order subgradient sampling avoids artificial critical points with probability one and applies moreover to a large range of risk minimization problems in deep learning, based on the backpropagation oracle. As byproducts of our approach, we obtain several results on integration of independent interest, such as an interchange result for conservative derivatives and integrals or the definability of set-valued parameterized integrals.