  Policy gradient methods are a widely used class of model-free reinforcement
learning algorithms where a state-dependent baseline is used to reduce gradient
estimator variance. Several recent papers extend the baseline to depend on both
the state and action and suggest that this significantly reduces variance and
improves sample efficiency without introducing bias into the gradient
estimates. To better understand this development, we decompose the variance of
the policy gradient estimator and numerically show that learned
state-action-dependent baselines do not in fact reduce variance over a
state-dependent baseline in commonly tested benchmark domains. We confirm this
unexpected result by reviewing the open-source code accompanying these prior
papers, and show that subtle implementation decisions cause deviations from the
methods presented in the papers and explain the source of the previously
observed empirical gains. Furthermore, the variance decomposition highlights
areas for improvement, which we demonstrate by illustrating a simple change to
the typical value function parameterization that can significantly improve
performance.
