A tutorial on automatic differentiation with complex numbers

TL;DR

This paper provides a comprehensive tutorial on automatic differentiation with complex numbers, emphasizing practical implementation and avoiding overly restrictive assumptions like holomorphicity, to aid users and developers in deriving custom gradients.

Contribution

It offers a detailed survey and implementation guide for forward- and reverse-mode automatic differentiation in complex numbers, focusing on Wirtinger derivatives and linear algebra techniques.

Findings

Derivation of complex Jacobian-vector and vector-Jacobian products using linear algebra.

Explicit avoidance of holomorphicity and Cauchy--Riemann equations in complex differentiation.

Guidance for implementing custom gradient rules for complex-valued functions.

Abstract

Automatic differentiation is everywhere, but there exists only minimal documentation of how it works in complex arithmetic beyond stating "derivatives in $\mathbb{C}^d$" $\cong$ "derivatives in $\mathbb{R}^{2d}$" and, at best, shallow references to Wirtinger calculus. Unfortunately, the equivalence $\mathbb{C}^d \cong \mathbb{R}^{2d}$ becomes insufficient as soon as we need to derive custom gradient rules, e.g., to avoid differentiating "through" expensive linear algebra functions or differential equation simulators. To combat such a lack of documentation, this article surveys forward- and reverse-mode automatic differentiation with complex numbers, covering topics such as Wirtinger derivatives, a modified chain rule, and different gradient conventions while explicitly avoiding holomorphicity and the Cauchy--Riemann equations (which would be far too restrictive). To be precise, we will derive, explain, and implement a complex version of Jacobian-vector and vector-Jacobian products almost entirely with linear algebra without relying on complex analysis or differential geometry. This tutorial is a call to action, for users and developers alike, to take complex values seriously when implementing custom gradient propagation rules -- the manuscript explains how.