  We show that several major algorithms of reinforcement learning (RL) fit into
the framework of categorical cybernetics, that is to say, parametrised
bidirectional processes. We build on our previous work in which we show that
value iteration can be represented by precomposition with a certain optic. The
outline of the main construction in this paper is: (1) We extend the Bellman
operators to parametrised optics that apply to action-value functions and
depend on a sample. (2) We apply a representable contravariant functor,
obtaining a parametrised function that applies the Bellman iteration. (3) This
parametrised function becomes the backward pass of another parametrised optic
that represents the model, which interacts with an environment via an agent.
Thus, parametrised optics appear in two different ways in our construction,
with one becoming part of the other. As we show, many of the major classes of
algorithms in RL can be seen as different extremal cases of this general setup:
dynamic programming, Monte Carlo methods, temporal difference learning, and
deep RL. We see this as strong evidence that this approach is a natural one and
believe that it will be a fruitful way to think about RL in the future.
