  General Value Function (GVF) is a powerful tool to represent both the {\em
predictive} and {\em retrospective} knowledge in reinforcement learning (RL).
In practice, often multiple interrelated GVFs need to be evaluated jointly with
pre-collected off-policy samples. In the literature, the gradient temporal
difference (GTD) learning method has been adopted to evaluate GVFs in the
off-policy setting, but such an approach may suffer from a large estimation
error even if the function approximation class is sufficiently expressive.
Moreover, none of the previous work have formally established the convergence
guarantee to the ground truth GVFs under the function approximation settings.
In this paper, we address both issues through the lens of a class of GVFs with
causal filtering, which cover a wide range of RL applications such as reward
variance, value gradient, cost in anomaly detection, stationary distribution
gradient, etc. We propose a new algorithm called GenTD for off-policy GVFs
evaluation and show that GenTD learns multiple interrelated multi-dimensional
GVFs as efficiently as a single canonical scalar value function. We further
show that unlike GTD, the learned GVFs by GenTD are guaranteed to converge to
the ground truth GVFs as long as the function approximation power is
sufficiently large. To our best knowledge, GenTD is the first off-policy GVF
evaluation algorithm that has global optimality guarantee.
