Position: Categorical Deep Learning is an Algebraic Theory of All Architectures

TL;DR

This paper advocates for a unified algebraic framework based on category theory to describe and analyze all deep learning architectures, bridging the gap between constraints and implementations.

Contribution

It introduces a novel categorical algebraic theory using monads in a 2-category to unify neural network constraints and implementations.

Findings

Recovers geometric deep learning constraints

Models diverse architectures like RNNs

Encodes standard CS and automata constructs

Abstract

We present our position on the elusive quest for a general-purpose framework for specifying and studying deep learning architectures. Our opinion is that the key attempts made so far lack a coherent bridge between specifying constraints which models must satisfy and specifying their implementations. Focusing on building a such a bridge, we propose to apply category theory -- precisely, the universal algebra of monads valued in a 2-category of parametric maps -- as a single theory elegantly subsuming both of these flavours of neural network design. To defend our position, we show how this theory recovers constraints induced by geometric deep learning, as well as implementations of many architectures drawn from the diverse landscape of neural networks, such as RNNs. We also illustrate how the theory naturally encodes many standard constructs in computer science and automata theory.