Reverse derivative categories

TL;DR

This paper axiomatizes categories with reverse derivatives, revealing that such categories inherently include forward derivatives with additional dagger structure, and characterizes the linear maps as additively enriched with dagger biproducts.

Contribution

It provides a direct axiomatization of reverse derivatives in categories, showing their relationship to forward derivatives and the structure of linear maps.

Findings

Reverse derivatives imply forward derivatives with dagger structure.

Linear maps form an additively enriched category with dagger biproducts.

Categories with reverse derivatives have a richer structure than those with only forward derivatives.

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.