Wasserstein policy gradient (WPG) updates state-conditional action laws by transport in the action space. We study entropy-regularized discounted linear-quadratic (LQ) control. A Bellman verification argument shows that the unrestricted problem has a linear-Gaussian optimal policy, and the discounted-occupancy-weighted statewise Wasserstein gradient is tangent to this policy class. WPG therefore reduces exactly to a finite-dimensional ODE for the feedback gain and action covariance. We prove that this ODE is globally well posed and converges exponentially from every admissible initialization. For each fixed LQ problem, the exponent has a positive limit as the entropy temperature tends to zero and contains no perturbative factor of the form $\exp(-c/τ)$, while retaining the usual dependence on the conditioning of the control problem.
When a reinforcement learning agent cannot observe the full state, we usually blame its policies: it cannot see enough to represent a good one. We show that in a solvable case the bigger problem lies elsewhere. Even when a good policy is available and the agent's value function is expressive enough to describe it exactly, learning still ends up somewhere far worse. We study a partially observed linear-quadratic problem in which a standard actor-critic learner can be solved in closed form. At our default setting the best policy the agent can represent is already close to optimal, costing 10.4% more than the ideal controller that observes everything. Learning does not find it. The algorithm instead comes to rest at a policy that is 35% worse than the best one available to it, and we can say exactly where and why. The cause is a bias in what the critic learns rather than a limit on what the actor can express. Because the agent cannot attribute what it sees to the part of the state it cannot observe, the critic misreads that unexplained variation as sharp curvature in its own value estimates, and the actor follows that error away from the optimum. We derive closed-form expressions for the resulting policy, for its cost, and for the one design choice that removes the problem, which is how far the learner looks ahead before trusting its own value estimates. Deep reinforcement learning experiments follow these predictions closely. Notably, giving the agent memory of past observations does not help, while changing how far it looks ahead does.