  Off-policy Reinforcement Learning (RL) holds the promise of better data
efficiency as it allows sample reuse and potentially enables safe interaction
with the environment. Current off-policy policy gradient methods either suffer
from high bias or high variance, delivering often unreliable estimates. The
price of inefficiency becomes evident in real-world scenarios such as
interaction-driven robot learning, where the success of RL has been rather
limited, and a very high sample cost hinders straightforward application. In
this paper, we propose a nonparametric Bellman equation, which can be solved in
closed form. The solution is differentiable w.r.t the policy parameters and
gives access to an estimation of the policy gradient. In this way, we avoid the
high variance of importance sampling approaches, and the high bias of
semi-gradient methods. We empirically analyze the quality of our gradient
estimate against state-of-the-art methods, and show that it outperforms the
baselines in terms of sample efficiency on classical control tasks.
