  While often stated as an instance of the likelihood ratio trick [Rubinstein,
1989], the original policy gradient theorem [Sutton, 1999] involves an integral
over the action space. When this integral can be computed, the resulting
"all-action" estimator [Sutton, 2001] provides a conditioning effect [Bratley,
1987] reducing the variance significantly compared to the REINFORCE estimator
[Williams, 1992]. In this paper, we adopt a numerical integration perspective
to broaden the applicability of the all-action estimator to general spaces and
to any function class for the policy or critic components, beyond the Gaussian
case considered by [Ciosek, 2018]. In addition, we provide a new theoretical
result on the effect of using a biased critic which offers more guidance than
the previous "compatible features" condition of [Sutton, 1999]. We demonstrate
the benefit of our approach in continuous control tasks with nonlinear function
approximation. Our results show improved performance and sample efficiency.
