  After presenting Actor Critic Methods (ACM), we show ACM are control variate
estimators. Using the projection theorem, we prove that the Q and Advantage
Actor Critic (A2C) methods are optimal in the sense of the $L^2$ norm for the
control variate estimators spanned by functions conditioned by the current
state and action. This straightforward application of Pythagoras theorem
provides a theoretical justification of the strong performance of QAC and AAC
most often referred to as A2C methods in deep policy gradient methods. This
enables us to derive a new formulation for Advantage Actor Critic methods that
has lower variance and improves the traditional A2C method.
