Paper:
The field of Deep Reinforcement Learning (DRL) has recently seen a surge in research in batch reinforcement learning, which aims for sample-efficient learning from a given data set without additional interactions with the environment. In the batch DRL setting, commonly employed off-policy DRL algorithms can perform poorly and sometimes even fail to learn altogether. In this paper we propose a new algorithm, Best-Action Imitation Learning (BAIL), which unlike many offpolicy DRL algorithms does not involve maximizing Q functions over the action space. Striving for simplicity as well as performance, BAIL first selects from the batch the actions it believes to be high-performing actions for their corresponding states; it then uses those state-action pairs to train a policy network using imitation learning. Although BAIL is simple, we demonstrate that BAIL achieves state of the art performance on the Mujoco benchmark.

Review:
Summary:
This paper studies the problem of learning a policy from a fixed dataset. The authors propose to estimate a smooth upper envelope of the episodic returns from the dataset as a state-value function. The policy is then learned by imitating the state action pairs from the dataset whose actual episodic return is close to the estimated envelope.
Recommended decision:
The direction of imitating "good" actions from the dataset is interesting. The intuition of estimating an upper envelope of the value function seems reasonable. However, I feel like this paper is not ready to be published in terms of its overall quality, mainly due to the lack of correctness, rigorousness and justification in statements and approaches.
Major comments:
- On the top of page 4:  "Because the Mujoco environments are continuing tasks, it is desirable to approximate the return over the infinite horizon, particularly for i values that are close to the (artificial) end of an episode. To do this, we note that the data-generation policy from one episode to the next typically changes slowly. We therefore apply a simple augmentation heuristic of concatenating the subsequent episode to the current episode, and running the sum in (1) to infinity." I cannot see how this approach is validated. The reset of initial state makes cross-episode cumulative reward from a state s not an approximation to the real return from state s. Estimating the infinite horizon return from finite horizon data is indeed a challenge here and simply cut the return at the end of an episode is be problematic. But the solution proposed by the authors is wrong in principle and cannot be simply justified by "good empirical performance". I feel hard to regard this choice a valid part of an algorithm unless further justification can be provided.
- Statements of theorems (4.1 and 4.2) are non-rigorous and contain irrelevant information: "lambda-smooth" is not an appropriate terminology when lambda is the weight of the regularizer. The actual "smoothness" also depends on the other term in the loss (same lambda does not indicate same smoothness in different objectives). For the same reason, Theorem 4.2 is wrong as changing K also changes the smoothness of the learned function. Proof of Theorem 4.2 in appendix is wrong as the authors ignore the coefficients in the last equation. Theorem 4.1-(1) cannot be true unless how V_\phi is parameterized is given: e.g. if there is no bias term or the regularization is applies to the bias term V will always output 0 as lambda 0-> \infty. The "2m+d" in Theorem 4.1-(2) is irrelevant to this work and cannot be justified without more detailed statements about how the network is parameterized. I appreciate the motivation that the authors try to validate the use of their objective to learn a "smooth upper envelope" but most of these statements are somewhat trivial and/or wrong section 4.1 does not actually deliver a valid justification.
- The use of "smooth upper envelope" itself can bring both over-estimation and under-estimation. For example, if one can concatenate different parts from different episodes to get a trajectory with higher return, the episodic return for the states along this trajectory is an under-estimate. Although it is fine to use a conservative estimate it would be better to be explicit about this and explain why this may not be a concern. On the other hand, it can bring over estimation to the state-values due to the smoothness enhanced to the fitted V. It would be better to see e.g. when these concerns do not matter (theoretically) or they are not real concerns in practice (by further inspecting the experiments). 
- Regarding Experiments: Why Hopper, Walker, HalfCheetah are trained with DDPG while Ant is trained by SAC? The performance of Final-DDPG/SAC after training for 1m steps looks way below what SAC and TD3 can get. Is it because they are just partially trained or noise is added to them? The baseline online-trained policy should not contain noise for a fair comparison. That said, in batch RL setting it is not necessary to compare to online-trained policy because it is a different setting. But if the authors want to compare to those, choice of baseline should be careful. An important baseline which is missing is to run vanilla DDPG/TD3/SAC as a batch-mode algorithm.  
Minor comments:
- Section 3, first paragraph: It is not very meaningful to say "simulators are deterministic so deterministic environments are important". Simulators are made by humans so they can be either deterministic or stochastic. "many robotic tasks are expected to be deterministic environments" is probably not true. I do not view "assuming deterministic envs" as a major limitation but I do not find these statements convincing as well. Similarly, the argument for studying non-stationary policy seems unsupportive: if the dataset comes from training a policy online then why do we care about learning another offline policy rather than just use or continue training the online policy. One argument I can see is that the online policy is worse. But the fact that these policies are worst than running e.g. SAC for a million steps makes the motivation questionable. Again, I do not view "choice of setting" as a limitation but I just find these statements a bit unsupportive. 
Potential directions for improvement:
To me the main part of the paper that looks problematic is Section 4.1 (both the approximation of infinite horizon returns and the theorems). It would be better to see a more rigorous and coherent justification of this approach (or some improved version), e.g. by either presenting analysis that is rigorous, correct and actually relevant or leave the space for more detailed empirical justification (e.g. whether potential over/under-estimating happens or not, comparing the estimated V to real episodic return of the learned policy).