On the linear convergence of policy gradient under Hadamard parameterization

The convergence of deterministic policy gradient under the Hadamard parameterization is studied in the tabular setting and the linear convergence of the algorithm is established. To this end, we first show that the error decreases at an $O(\frac{1}{k})$ rate for all the iterations. Based on this result, we further show that the algorithm has a faster local linear convergence rate after $k_{0}$ iterations, where $k_{0}$ is a constant that only depends on the MDP problem and the initialization. To show the local linear convergence of the algorithm, we have indeed established the contraction of the sub-optimal probability $b_{s}<^>{k}$ (i.e. the probability of the output policy $\pi <^>{k}$ on non-optimal actions) when $k\ge k_{0}$.