# Conservative and semismooth derivatives are equivalent for semialgebraic maps

arXiv: https://arxiv.org/abs/2102.08484 · HTML: https://mlsanity.com/paper/2102.08484/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Feb 2021.

## 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)
- [Path differentiability of ODE flows](https://mlsanity.com/paper/2201.03819/) (2022)
- [Efficient and provably convergent end-to-end training of deep neural networks with linear constraints](https://mlsanity.com/paper/2605.11526/) (2026)

## Abstract (arXiv)

Subgradient and Newton algorithms for nonsmooth optimization require generalized derivatives to satisfy subtle approximation properties: conservativity for the former and semismoothness for the latter. Though these two properties originate in entirely different contexts, we show that in the semi-algebraic setting they are equivalent. Both properties for a generalized derivative simply require it to coincide with the standard directional derivative on the tangent spaces of some partition of the domain into smooth manifolds. An appealing byproduct is a new short proof that semi-algebraic maps are semismooth relative to the Clarke Jacobian.