A Hitchhiker's Guide to Automatic Differentiation

TL;DR

This paper provides a comprehensive overview of the mathematical foundations and various implementations of Automatic Differentiation, including forward and reverse modes, highlighting their equivalences and differences.

Contribution

It offers a unified mathematical perspective on different AD methods, clarifying their relationships and explaining higher-order extensions.

Findings

Different descriptions of Forward Mode AD are mathematically equivalent.

The paper clarifies the relationship between various AD implementations.

It briefly introduces the Reverse Mode of AD.

Abstract

This article provides an overview of some of the mathematical principles of Automatic Differentiation (AD). In particular, we summarise different descriptions of the Forward Mode of AD, like the matrix-vector product based approach, the idea of lifting functions to the algebra of dual numbers, the method of Taylor series expansion on dual numbers and the application of the push-forward operator, and explain why they all reduce to the same actual chain of computations. We further give a short mathematical description of some methods of higher-order Forward AD and, at the end of this paper, briefly describe the Reverse Mode of Automatic Differentiation.