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A supervised learning algorithm searches over a set of functions $A \to B$parametrised by a space $P$ to find the best approximation to some idealfunction $f\colon A \to B$. It does this by taking examples $(a,f(a)) \inA\times B$, and updating the parameter according to some rule. We define acategory where these update rules may be composed, and show that gradientdescent---with respect to a fixed step size and an error function satisfying acertain property---defines a monoidal functor from a category of parametrisedfunctions to this category of update rules. This provides a structuralperspective on backpropagation, as well as a broad generalisation of neuralnetworks.

Backprop as Functor: A compositional perspective on supervised learning