Compositionality of optimal control laws

We present a theory of compositionality in stochastic optimal control, showing how task-optimal controllers can be constructed from certain primitives. The primitives are themselves feedback controllers pursuing their own agendas. They are mixed in proportion to how much progress they are making towards their agendas and how compatible their agendas are with the present task. The resulting composite control law is provably optimal when the problem belongs to a certain class. This class is rather general and yet has a number of unique properties - one of which is that the Bellman equation can be made linear even for non-linear or discrete dynamics. This gives rise to the compositionality developed here. In the special case of linear dynamics and Gaussian noise our framework yields analytical solutions (i.e. non-linear mixtures of linear-quadratic regulators) without requiring the final cost to be quadratic. More generally, a natural set of control primitives can be constructed by applying SVD to Greens function of the Bellman equation. We illustrate the theory in the context of human arm movements. The ideas of optimality and compositionality are both very prominent in the field of motor control, yet they are hard to reconcile. Our work makes this possible.