  Many reinforcement learning algorithms rely on value estimation, however, the
most widely used algorithms -- namely temporal difference algorithms -- can
diverge under both off-policy sampling and nonlinear function approximation.
Many algorithms have been developed for off-policy value estimation based on
the linear mean squared projected Bellman error (MSPBE) and are sound under
linear function approximation. Extending these methods to the nonlinear case
has been largely unsuccessful. Recently, several methods have been introduced
that approximate a different objective -- the mean-squared Bellman error (MSBE)
-- which naturally facilitate nonlinear approximation. In this work, we build
on these insights and introduce a new generalized MSPBE that extends the linear
MSPBE to the nonlinear setting. We show how this generalized objective unifies
previous work and obtain new bounds for the value error of the solutions of the
generalized objective. We derive an easy-to-use, but sound, algorithm to
minimize the generalized objective, and show that it is more stable across
runs, is less sensitive to hyperparameters, and performs favorably across four
control domains with neural network function approximation.
