Reverse derivative categories

Robin Cockett; Geoffrey Cruttwell; Jonathan Gallagher; Jean-Simon Pacaud Lemay; Benjamin MacAdam; Gordon Plotkin; Dorette Pronk

arXiv · 1910.07065

Reverse derivative categories

Robin Cockett·Geoffrey Cruttwell·Jonathan Gallagher·Jean-Simon Pacaud Lemay·Benjamin MacAdam·Gordon Plotkin·Dorette Pronk

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Abstract

The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.

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Robin Cockett, Geoffrey Cruttwell, Jonathan Gallagher, Jean-Simon Pacaud Lemay, Benjamin MacAdam, Gordon Plotkin, Dorette Pronk. 2019-10-15. Reverse derivative categories. https://arxiv.org/abs/1910.07065

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