  To accumulate knowledge and improve its policy of behaviour, a reinforcement
learning agent can learn `off-policy' about policies that differ from the
policy used to generate its experience. This is important to learn
counterfactuals, or because the experience was generated out of its own
control. However, off-policy learning is non-trivial, and standard
reinforcement-learning algorithms can be unstable and divergent.
  In this paper we discuss a novel family of off-policy prediction algorithms
which are convergent by construction. The idea is to first learn on-policy
about the data-generating behaviour, and then bootstrap an off-policy value
estimate on this on-policy estimate, thereby constructing a value estimate that
is partially off-policy. This process can be repeated to build a chain of value
functions, each time bootstrapping a new estimate on the previous estimate in
the chain. Each step in the chain is stable and hence the complete algorithm is
guaranteed to be stable. Under mild conditions this comes arbitrarily close to
the off-policy TD solution when we increase the length of the chain. Hence it
can compute the solution even in cases where off-policy TD diverges.
  We prove that the proposed scheme is convergent and corresponds to an
iterative decomposition of the inverse key matrix. Furthermore it can be
interpreted as estimating a novel objective -- that we call a `k-step
expedition' -- of following the target policy for finitely many steps before
continuing indefinitely with the behaviour policy. Empirically we evaluate the
idea on challenging MDPs such as Baird's counter example and observe favourable
results.
