  We revisit the estimation bias in policy gradients for the discounted
episodic Markov decision process (MDP) from Deep Reinforcement Learning (DRL)
perspective. The objective is formulated theoretically as the expected returns
discounted over the time horizon. One of the major policy gradient biases is
the state distribution shift: the state distribution used to estimate the
gradients differs from the theoretical formulation in that it does not take
into account the discount factor. Existing discussion of the influence of this
bias was limited to the tabular and softmax cases in the literature. Therefore,
in this paper, we extend it to the DRL setting where the policy is
parameterized and demonstrate how this bias can lead to suboptimal policies
theoretically. We then discuss why the empirically inaccurate implementations
with shifted state distribution can still be effective. We show that, despite
such state distribution shift, the policy gradient estimation bias can be
reduced in the following three ways: 1) a small learning rate; 2) an
adaptive-learning-rate-based optimizer; and 3) KL regularization. Specifically,
we show that a smaller learning rate, or, an adaptive learning rate, such as
that used by Adam and RSMProp optimizers, makes the policy optimization robust
to the bias. We further draw connections between optimizers and the
optimization regularization to show that both the KL and the reverse KL
regularization can significantly rectify this bias. Moreover, we provide
extensive experiments on continuous control tasks to support our analysis. Our
paper sheds light on how successful PG algorithms optimize policies in the DRL
setting, and contributes insights into the practical issues in DRL.
