Learners' Languages

TL;DR

This paper explores the categorical structures underlying deep learning, connecting gradient descent and backpropagation to polynomial functors and dynamical systems, and discusses their logical and theoretical implications.

Contribution

It demonstrates that categories of learners relate to polynomial functors and dynamical systems, providing a new categorical perspective on deep learning mechanisms.

Findings

Slens is a full subcategory of Poly, linking lenses to polynomial functors.

Maps in Para(Slens) correspond to generalized Moore machines.

The category p-Coalg forms a topos, enabling logical reasoning about dynamical systems.

Abstract

In "Backprop as functor", the authors show that the fundamental elements of deep learning -- gradient descent and backpropagation -- can be conceptualized as a strong monoidal functor Para(Euc)$\to$Learn from the category of parameterized Euclidean spaces to that of learners, a category developed explicitly to capture parameter update and backpropagation. It was soon realized that there is an isomorphism Learn$\cong$Para(Slens), where Slens is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that Slens is a full subcategory of Poly, the category of polynomial functors in one variable, via the functor $A\mapsto Ay^A$. Using the fact that (Poly,$\otimes$) is monoidal closed, we show that a map $A\to B$ in Para(Slens) has a natural interpretation in terms of dynamical systems (more precisely, generalized Moore machines) whose interface is the internal-hom type $[Ay^A,By^B]$. Finally, we review the fact that the category p-Coalg of dynamical systems on any $p \in$ Poly forms a topos, and consider the logical propositions that can be stated in its internal language. We give gradient descent as an example, and we conclude by discussing some directions for future work.