  Off-policy learning is key to scaling up reinforcement learning as it allows
to learn about a target policy from the experience generated by a different
behavior policy. Unfortunately, it has been challenging to combine off-policy
learning with function approximation and multi-step bootstrapping in a way that
leads to both stable and efficient algorithms. In this work, we show that the
\textsc{Tree Backup} and \textsc{Retrace} algorithms are unstable with linear
function approximation, both in theory and in practice with specific examples.
Based on our analysis, we then derive stable and efficient gradient-based
algorithms using a quadratic convex-concave saddle-point formulation. By
exploiting the problem structure proper to these algorithms, we are able to
provide convergence guarantees and finite-sample bounds. The applicability of
our new analysis also goes beyond \textsc{Tree Backup} and \textsc{Retrace} and
allows us to provide new convergence rates for the GTD and GTD2 algorithms
without having recourse to projections or Polyak averaging.
