Combining fixpoint and differentiation theory

TL;DR

This paper develops a categorical framework combining fixpoints and derivatives, enabling formal reasoning about their interactions and applications like Newton-Raphson optimization in higher order languages.

Contribution

It introduces an axiom linking derivatives of fixpoints with fixpoints of derivatives within Cartesian differential categories, expanding the theoretical understanding of their interplay.

Findings

Canonical models of the combined notion are provided.

The framework formalizes Newton-Raphson optimization for fixpoints.

Extensions to higher order languages are demonstrated.

Abstract

Interactions between derivatives and fixpoints have many important applications in both computer science and mathematics. In this paper, we provide a categorical framework to combine fixpoints with derivatives by studying Cartesian differential categories with a fixpoint operator. We introduce an additional axiom relating the derivative of a fixpoint with the fixpoint of the derivative. We show how the standard examples of Cartesian differential categories where we can compute fixpoints provide canonical models of this notion. We also consider when the fixpoint operator is a Conway operator, or when the underlying category is closed. As an application, we show how this framework is a suitable setting to formalize the Newton-Raphson optimization for fast approximation of fixpoints and extend it to higher order languages.