# The structure of conservative gradient fields

arXiv: https://arxiv.org/abs/2101.00699 · HTML: https://mlsanity.com/paper/2101.00699/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 13 references that are in the map.
- First posted on arXiv 3 Jan 2021.

## Builds on (most cited first, 4 of 4)

- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [A mathematical model for automatic differentiation in machine learning](https://mlsanity.com/paper/2006.02080/) (2020)
- [On Correctness of Automatic Differentiation for Non-Differentiable Functions](https://mlsanity.com/paper/2006.06903/) (2020)

## Built on by (most cited first, 4 of 4)

- [Nonsmooth Implicit Differentiation for Machine Learning and Optimization](https://mlsanity.com/paper/2106.04350/) (2021)
- [On the complexity of nonsmooth automatic differentiation](https://mlsanity.com/paper/2206.01730/) (2022)
- [Deep Learning as the Disciplined Construction of Tame Objects](https://mlsanity.com/paper/2509.18025/) (2025)
- [Nonsmooth Optimization via Orthogonalized Momentum](https://mlsanity.com/paper/2609.13677/) (2026)

## Abstract (arXiv)

The classical Clarke subdifferential alone is inadequate for understanding automatic differentiation in nonsmooth contexts. Instead, we can sometimes rely on enlarged generalized gradients called "conservative fields", defined through the natural path-wise chain rule: one application is the convergence analysis of gradient-based deep learning algorithms. In the semi-algebraic case, we show that all conservative fields are in fact just Clarke subdifferentials plus normals of manifolds in underlying Whitney stratifications.