  Our goal is to enable robots to learn cost functions from user guidance.
Often it is difficult or impossible for users to provide full demonstrations,
so corrections have emerged as an easier guidance channel. However, when robots
learn cost functions from corrections rather than demonstrations, they have to
extrapolate a small amount of information -- the change of a waypoint along the
way -- to the rest of the trajectory. We cast this extrapolation problem as
online function approximation, which exposes different ways in which the robot
can interpret what trajectory the person intended, depending on the function
space used for the approximation. Our simulation results and user study suggest
that using function spaces with non-Euclidean norms can better capture what
users intend, particularly if environments are uncluttered. This, in turn, can
lead to the robot learning a more accurate cost function and improves the
user's subjective perceptions of the robot.
