  We consider off-policy temporal-difference (TD) learning in discounted Markov
decision processes, where the goal is to evaluate a policy in a model-free way
by using observations of a state process generated without executing the
policy. To curb the high variance issue in off-policy TD learning, we propose a
new scheme of setting the $\lambda$-parameters of TD, based on generalized
Bellman equations. Our scheme is to set $\lambda$ according to the eligibility
trace iterates calculated in TD, thereby easily keeping these traces in a
desired bounded range. Compared with prior work, this scheme is more direct and
flexible, and allows much larger $\lambda$ values for off-policy TD learning
with bounded traces. As to its soundness, using Markov chain theory, we prove
the ergodicity of the joint state-trace process under nonrestrictive
conditions, and we show that associated with our scheme is a generalized
Bellman equation (for the policy to be evaluated) that depends on both the
evolution of $\lambda$ and the unique invariant probability measure of the
state-trace process. These results not only lead immediately to a
characterization of the convergence behavior of least-squares based
implementation of our scheme, but also prepare the ground for further analysis
of gradient-based implementations.
