A Functional Framework for Nonsmooth Autodiff with {\it Maxpooling} Functions
Abstract
We make a comment on the recent work by Boustany, by showing that the Murat-TrombettiTheorem provides a simple and efficient mathematical framework for nonsmooth automatic differentiation of {\it maxpooling} functions. In particular it gives a the chain rule formula which correctly defines the composition of Lipschitz-continuous functions which are piecewise $C^1$. The formalism is applied to four basic examples, with some tests in PyTorch. A self contained proof of an important Stampacchia formula is in the appendix.
Cite
Text
Després. "A Functional Framework for Nonsmooth Autodiff with {\it Maxpooling} Functions." Transactions on Machine Learning Research, 2025.Markdown
[Després. "A Functional Framework for Nonsmooth Autodiff with {\it Maxpooling} Functions." Transactions on Machine Learning Research, 2025.](https://mlanthology.org/tmlr/2025/despres2025tmlr-functional/)BibTeX
@article{despres2025tmlr-functional,
title = {{A Functional Framework for Nonsmooth Autodiff with {\it Maxpooling} Functions}},
author = {Després, Bruno},
journal = {Transactions on Machine Learning Research},
year = {2025},
url = {https://mlanthology.org/tmlr/2025/despres2025tmlr-functional/}
}