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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 trajectoryprediction and optimization. Interestingly, most all prior work in imitationand reinforcement learning builds on a sequential trajectory representation --calculating the next state in the trajectory given its predecessors. We proposea different perspective: a goal-conditioned trajectory can be represented byfirst selecting an intermediate state between start and goal, partitioning thetrajectory into two. Then, recursively, predicting intermediate points on eachsub-segment, until a complete trajectory is obtained. We call thisrepresentation a sub-goal tree, and building on it, we develop new methods fortrajectory prediction, learning, and optimization. We show that in a supervisedlearning setting, sub-goal trees better account for trajectory variability, andcan predict trajectories exponentially faster at test time by leveraging aconcurrent computation. Then, for optimization, we derive a new dynamicprogramming equation for sub-goal trees, and use it to develop new planning andreinforcement learning algorithms. These algorithms, which are not based on thestandard Bellman equation, naturally account for hierarchical sub-goalstructure in a task. Empirical results on motion planning domains show that thesub-goal tree framework significantly improves both accuracy and predictiontime.

Sub-Goal Trees -- a Framework for Goal-Directed Trajectory Prediction and Optimization