  A temporally abstract action, or an option, is specified by a policy and a
termination condition: the policy guides option behavior, and the termination
condition roughly determines its length. Generally, learning with longer
options (like learning with multi-step returns) is known to be more efficient.
However, if the option set for the task is not ideal, and cannot express the
primitive optimal policy exactly, shorter options offer more flexibility and
can yield a better solution. Thus, the termination condition puts learning
efficiency at odds with solution quality. We propose to resolve this dilemma by
decoupling the behavior and target terminations, just like it is done with
policies in off-policy learning. To this end, we give a new algorithm,
Q(\beta), that learns the solution with respect to any termination condition,
regardless of how the options actually terminate. We derive Q(\beta) by casting
learning with options into a common framework with well-studied multi-step
off-policy learning. We validate our algorithm empirically, and show that it
holds up to its motivating claims.
