  We consider off-policy policy evaluation with function approximation (FA) in
average-reward MDPs, where the goal is to estimate both the reward rate and the
differential value function. For this problem, bootstrapping is necessary and,
along with off-policy learning and FA, results in the deadly triad (Sutton &
Barto, 2018). To address the deadly triad, we propose two novel algorithms,
reproducing the celebrated success of Gradient TD algorithms in the
average-reward setting. In terms of estimating the differential value function,
the algorithms are the first convergent off-policy linear function
approximation algorithms. In terms of estimating the reward rate, the
algorithms are the first convergent off-policy linear function approximation
algorithms that do not require estimating the density ratio. We demonstrate
empirically the advantage of the proposed algorithms, as well as their
nonlinear variants, over a competitive density-ratio-based approach, in a
simple domain as well as challenging robot simulation tasks.
