Learners' Languages

2021/03/01 by Spivak, David I.
#Category Theory (math.CT) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG)

paper · doi:10.48550/arxiv.2103.01189

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)→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≅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↦ Ay^A. Using the fact that (Poly,⊗) is monoidal closed, we show that a map A→ 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 ∈ 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.

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