Dual Policy Iteration
A novel class of Approximate Policy Iteration (API) algorithms have recently demonstrated impressive practical performance (e.g., ExIt [1], AlphaGo-Zero [2]). This new family of algorithms maintains, and alternately optimizes, two policies: a fast, reactive policy (e.g., a deep neural network) deployed at test time, and a slow, non-reactive policy (e.g., Tree Search), that can plan multiple steps ahead. The reactive policy is updated under supervision from the non-reactive policy, while the non-reactive policy is improved via guidance from the reactive policy. In this work we study this class of Dual Policy Iteration (DPI) strategy in an alternating optimization framework and provide a convergence analysis that extends existing API theory. We also develop a special instance of this framework which reduces the update of non-reactive policies to model-based optimal control using learned local models, and provides a theoretically sound way of unifying model-free and model-based RL approaches with unknown dynamics. We demonstrate the efficacy of our approach on various continuous control Markov Decision Processes.
Major Comments:  1) page 3: How do you set a parameter $\alpha$ and what effect it has? In other words, if $\eta^* = \argmin J(\eta)$ and $\eta_{\alpha}$ is the solution of Eq , what can you say about relation between $\eta^*$ and $\eta_{\alpha}$?  2) page 3: parameter $\delta$ in Eq 3. Could you please elaborate more on how are you going to choose? If it does depend on policy $\pi_n$, then we need to replace it by $\delta_n$ (i.e. it is changing from iteration to iteration). Also, apparently, this parameter will have an effect on the transition policy $\hat{P}$ that is found, and, hence, has effect to the policy $\eta_n$ evaluated at iteration $n$. Did you study this effect?  3) page 3: Theorem 3.1. I am confused with this result. Because the real transition $P$ is not available then we need to study the $J(\eta_n)$ evaluated with model $\hat{P}$ instead of $P$. But Theorem 3.1 studies the quality of policy $\eta_n$ with respect to real model $P$. Therefore, Theorem 3.1 doesn't quantify the quality of the tree search policy $\eta_n$.   4) page 4: Section 3.2. How do you define parameter $\beta$ in Eq 6. How effect does it have on the approximate solution $\pi_{n+1}$?  5) Due to PDL Lemma, for a given fast reactive policy $\pi_n$ the corresponding tree search policy $\eta_n$ actually doesn't depend on $\pi_n$. This is because policy $eta_n = argmin_{\eta} J(\eta)$, and $J(\eta)$ doesn’t depend on $pi_n$. The dependence appears when we impose trust region constraint given in terms of $\pi_n$.  My biggest issue with this paper is about the parameters $\alpha, \delta, \beta$ that was introduced as some form of relaxation but, at the same time, no effect of them were studied. (Please see my comment on Theorem 3.1 in 3) for better understanding).   Minor Comments: 1) page 3: Although the min-max framework considered in Section 3 is understandable, it would be good to elaborate on for readers.  Apart from the above major and minor comments, overall the paper appears to be written and structured well. Technically novel.