Learners' languages

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 $\mathbf{Para}(\mathbf{Euc})\to\mathbf{Learn}$ 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 $\mathbf{Learn}\cong\mathbf{Para}(\mathbf{SLens})$, where $\mathbf{SLens}$ is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that $\mathbf{SLens}$ is a full subcategory of $\mathbf{Poly}$, the category of polynomial functors in one variable, via the functor $A\mapsto Ay^A$. Using the fact that $(\mathbf{Poly},\otimes)$ is monoidal closed, we show that a map $A\to B$ in $\mathbf{Para}(\mathbf{SLens})$ 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 $p\text{-}\mathbf{Coalg}$ of dynamical systems on any $p\in\mathbf{Poly}$ 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.