  To estimate the value functions of policies from exploratory data, most
model-free off-policy algorithms rely on importance sampling, where the use of
importance sampling ratios often leads to estimates with severe variance. It is
thus desirable to learn off-policy without using the ratios. However, such an
algorithm does not exist for multi-step learning with function approximation.
In this paper, we introduce the first such algorithm based on
temporal-difference (TD) learning updates. We show that an explicit use of
importance sampling ratios can be eliminated by varying the amount of
bootstrapping in TD updates in an action-dependent manner. Our new algorithm
achieves stability using a two-timescale gradient-based TD update. A prior
algorithm based on lookup table representation called Tree Backup can also be
retrieved using action-dependent bootstrapping, becoming a special case of our
algorithm. In two challenging off-policy tasks, we demonstrate that our
algorithm is stable, effectively avoids the large variance issue, and can
perform substantially better than its state-of-the-art counterpart.
