  The policy gradient theorem is defined based on an objective with respect to
the initial distribution over states. In the discounted case, this results in
policies that are optimal for one distribution over initial states, but may not
be uniformly optimal for others, no matter where the agent starts from.
Furthermore, to obtain unbiased gradient estimates, the starting point of the
policy gradient estimator requires sampling states from a normalized discounted
weighting of states. However, the difficulty of estimating the normalized
discounted weighting of states, or the stationary state distribution, is quite
well-known. Additionally, the large sample complexity of policy gradient
methods is often attributed to insufficient exploration, and to remedy this, it
is often assumed that the restart distribution provides sufficient exploration
in these algorithms. In this work, we propose exploration in policy gradient
methods based on maximizing entropy of the discounted future state
distribution. The key contribution of our work includes providing a practically
feasible algorithm to estimate the normalized discounted weighting of states,
i.e, the \textit{discounted future state distribution}. We propose that
exploration can be achieved by entropy regularization with the discounted state
distribution in policy gradients, where a metric for maximal coverage of the
state space can be based on the entropy of the induced state distribution. The
proposed approach can be considered as a three time-scale algorithm and under
some mild technical conditions, we prove its convergence to a locally optimal
policy. Experimentally, we demonstrate usefulness of regularization with the
discounted future state distribution in terms of increased state space coverage
and faster learning on a range of complex tasks.
