  Previous work in hierarchical reinforcement learning has faced a dilemma:
either ignore the values of different possible exit states from a subroutine,
thereby risking suboptimal behavior, or represent those values explicitly
thereby incurring a possibly large representation cost because exit values
refer to nonlocal aspects of the world (i.e., all subsequent rewards). This
paper shows that, in many cases, one can avoid both of these problems. The
solution is based on recursively decomposing the exit value function in terms
of Q-functions at higher levels of the hierarchy. This leads to an intuitively
appealing runtime architecture in which a parent subroutine passes to its child
a value function on the exit states and the child reasons about how its choices
affect the exit value. We also identify structural conditions on the value
function and transition distributions that allow much more concise
representations of exit state distributions, leading to further state
abstraction. In essence, the only variables whose exit values need be
considered are those that the parent cares about and the child affects. We
demonstrate the utility of our algorithms on a series of increasingly complex
environments.
