  In this paper we discuss $\l$-policy iteration, a method for exact and
approximate dynamic programming. It is intermediate between the classical value
iteration (VI) and policy iteration (PI) methods, and it is closely related to
optimistic (also known as modified) PI, whereby each policy evaluation is done
approximately, using a finite number of VI. We review the theory of the method
and associated questions of bias and exploration arising in simulation-based
cost function approximation. We then discuss various implementations, which
offer advantages over well-established PI methods that use LSPE($\l$),
LSTD($\l$), or TD($\l$) for policy evaluation with cost function approximation.
One of these implementations is based on a new simulation scheme, called
geometric sampling, which uses multiple short trajectories rather than a single
infinitely long trajectory.
