Building upon the recent success of deep reinforcement learning methods, we investigate the possibility of on-policy reinforcement learning improvement by reusing the data from several consecutive policies. On-policy methods bring many benefits, such as ability to evaluate each resulting policy. However, they usually discard all the information about the policies which existed before. In this work, we propose adaptation of the replay buffer concept, borrowed from the off-policy learning setting, to create the method, combining advantages of on-and off-policy learning. To achieve this, the proposed algorithm generalises the Q-, value and advantage functions for data from multiple policies. The method uses trust region optimisation, while avoiding some of the common problems of the algorithms such as TRPO or ACKTR: it uses hyperparameters to replace the trust region selection heuristics, as well as the trainable covariance matrix instead of the fixed one. In many cases, the method not only improves the results comparing to the state-of-the-art trust region on-policy learning algorithms such as PPO, ACKTR and TRPO, but also with respect to their off-policy counterpart DDPG.
The authors introduce a off-policy method for TRPO by suggesting to use replay buffers to store trajectories and sample from them during training. To do this they extend the definition of the Q function to multiple policies where the Q_pi bar is then the expectation over the several policies. They propose the same for the value function and consequently the advantage function. 
In my opinion this is some interesting work, but there are some details that are not clear to me, so i have several questions.
1. Why is it necessary to define these generalized notions of the Q, Value and Advantage functions? You motivate this by the fact the samples stored in the replay buffer will be generated by different policies, i.e. by differently parametrized policies at a certain time step. But this also holds almost all algorithms using replay buffers. Could you plese explain this part further?
2. In eq. (26) you introduce the parameter alpha as a sort of Lagrange multiplier to turn the unconstrained optimization problem defined by TRPO into a constrained one. This is was also proposed early by Schulman et al. in Proximal Policy Optimization. Yet, it is not cited or referenced. In the discussion of the experimental results go further into this. Please explain this part in more detail.
3. Another point of your work is the learnable diagonal covariance matrix. How can you be sure that the improvements you show are due to the replay buffers and not due to learning these? Or learning covariance in combination with the penalty term alpha?
4. Can you provide comparative results for PPO? PPO outperforms DDP and TRPO on most tasks so it would be interessting to see
5. How many trajectory samples do you store in the replay buffers? Can you provide results where you use your method but without any replay buffers, i.e. by using the last batch of data points?
Minor Suggestions:
- The references for the figures in the Experiments part are off. In fig. 1 you cite Todorov et al. for Mujoco but not TRPO and ACKTR, the same in fig. 2. Then in fig. 3 you cite DDPG also with Todorov et al.
- Some parts of the text is a bit unorganized. In section 2.1 you introduce AC algorithms and on the next page you give the definitions for all components but you don't say anything about how the interact. Also, the definition of the expected return was not "invented" by Schulman et al, and neither were Advantages, Q-, and Value functions. Maybe add a second or third reference.