  Many AI problems, in robotics and other domains, are goal-directed,
essentially seeking a trajectory leading to some goal state. In such problems,
the way we choose to represent a trajectory underlies algorithms for trajectory
prediction and optimization. Interestingly, most all prior work in imitation
and reinforcement learning builds on a sequential trajectory representation --
calculating the next state in the trajectory given its predecessors. We propose
a different perspective: a goal-conditioned trajectory can be represented by
first selecting an intermediate state between start and goal, partitioning the
trajectory into two. Then, recursively, predicting intermediate points on each
sub-segment, until a complete trajectory is obtained. We call this
representation a sub-goal tree, and building on it, we develop new methods for
trajectory prediction, learning, and optimization. We show that in a supervised
learning setting, sub-goal trees better account for trajectory variability, and
can predict trajectories exponentially faster at test time by leveraging a
concurrent computation. Then, for optimization, we derive a new dynamic
programming equation for sub-goal trees, and use it to develop new planning and
reinforcement learning algorithms. These algorithms, which are not based on the
standard Bellman equation, naturally account for hierarchical sub-goal
structure in a task. Empirical results on motion planning domains show that the
sub-goal tree framework significantly improves both accuracy and prediction
time.
