  In this paper, we present a policy gradient method that avoids exploratory
noise injection and performs policy search over the deterministic landscape. By
avoiding noise injection all sources of estimation variance can be eliminated
in systems with deterministic dynamics (up to the initial state distribution).
Since deterministic policy regularization is impossible using traditional
non-metric measures such as the KL divergence, we derive a Wasserstein-based
quadratic model for our purposes. We state conditions on the system model under
which it is possible to establish a monotonic policy improvement guarantee,
propose a surrogate function for policy gradient estimation, and show that it
is possible to compute exact advantage estimates if both the state transition
model and the policy are deterministic. Finally, we describe two novel robotic
control environments -- one with non-local rewards in the frequency domain and
the other with a long horizon (8000 time-steps) -- for which our policy
gradient method (TDPO) significantly outperforms existing methods (PPO, TRPO,
DDPG, and TD3). Our implementation with all the experimental settings is
available at https://github.com/ehsansaleh/code_tdpo
