Backprop as Functor: A compositional perspective on supervised learning

A supervised learning algorithm searches over a set of functions $A\rightarrow B$ parametrised by a space $P$ to find the best approximation to some ideal function $f:A\rightarrow B$. It does this by taking examples $(a, f(a))\in A\times B$, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent-with respect to a fixed step size and an error function satisfying a certain property-defines a monoidal functor from a category of parametrised functions to this category of update rules. A key contribution is the notion of request function. This provides a structural perspective on backpropagation, giving a broad generalisation of neural networks and linking it with structures from bidirectional programming and open games.