  We prove that a single-layer neural network trained with the online actor
critic algorithm converges in distribution to a random ordinary differential
equation (ODE) as the number of hidden units and the number of training steps
$\rightarrow \infty$. In the online actor-critic algorithm, the distribution of
the data samples dynamically changes as the model is updated, which is a key
challenge for any convergence analysis. We establish the geometric ergodicity
of the data samples under a fixed actor policy. Then, using a Poisson equation,
we prove that the fluctuations of the model updates around the limit
distribution due to the randomly-arriving data samples vanish as the number of
parameter updates $\rightarrow \infty$. Using the Poisson equation and weak
convergence techniques, we prove that the actor neural network and critic
neural network converge to the solutions of a system of ODEs with random
initial conditions. Analysis of the limit ODE shows that the limit critic
network will converge to the true value function, which will provide the actor
an asymptotically unbiased estimate of the policy gradient. We then prove that
the limit actor network will converge to a stationary point.
