  The policy gradient theorem states that the policy should only be updated in
states that are visited by the current policy, which leads to insufficient
planning in the off-policy states, and thus to convergence to suboptimal
policies. We tackle this planning issue by extending the policy gradient theory
to policy updates with respect to any state density. Under these generalized
policy updates, we show convergence to optimality under a necessary and
sufficient condition on the updates' state densities, and thereby solve the
aforementioned planning issue. We also prove asymptotic convergence rates that
significantly improve those in the policy gradient literature.
  To implement the principles prescribed by our theory, we propose an agent, Dr
Jekyll & Mr Hyde (JH), with a double personality: Dr Jekyll purely exploits
while Mr Hyde purely explores. JH's independent policies allow to record two
separate replay buffers: one on-policy (Dr Jekyll's) and one off-policy (Mr
Hyde's), and therefore to update JH's models with a mixture of on-policy and
off-policy updates. More than an algorithm, JH defines principles for
actor-critic algorithms to satisfy the requirements we identify in our
analysis. We extensively test on finite MDPs where JH demonstrates a superior
ability to recover from converging to a suboptimal policy without impairing its
speed of convergence. We also implement a deep version of the algorithm and
test it on a simple problem where it shows promising results.
