__ Summary and Contributions__: #### Post author feedback ####
The authors address my concerns in their rebuttal. Therefore, I increase my score to 7.
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The paper studies off-policy policy evaluation in reinforcement learning with function approximation. It unifies two styles of learning “value learning and weight learning” via the introduced minimax confidence interval (CI) method and it characterizes the bias due to the approximation errors. Finally, the paper provides experimental results to support its theoretical findings in both policy evaluation and policy optimization setting.

__ Strengths__: The paper devises by simple derivations the minimax confidence interval approaches. It is closely connected to previous methods introduced in prior works such as MQL, MWL and AlgaeDICE but the paper does a good job to unify them. The author offer also insight on the behavior of CI in the misspecification setting (for both the marginalized importance weights and the Q-value) by characterizing the validity and the tightness of the CI.
The paper also conducted a set of experiments to illustrate some theoretical predictions.

__ Weaknesses__: The empirical part of the paper is limited. I can understand that maybe the focus of the papier is rather theoretical but it would be nice to improve the empirical part.
In particular, in section 4.4 it would be nice to see the effect of the expressive power of W on the the validity and the tightness of CI.
In Figure 1, could you also explain why we tend to have larger CI with large Q network and why the CI tend to exclude slightly the ground truth with moderately large Q class ?
In figure 3, the validity ratio is not defined before, how is it computed ?
For the policy optimization, in figure 2. MUB-PO fails totally. Authors explains this by the fact that MUB-PO may induce state distribution that is different from the training distribution. They also relate to MUB-PO to R_max, a classical algorithm designed for exploration.
It would be very great to run MUB-PO is online setting to assess its exploration capability as claimed by the paper.

__ Correctness__: I went through derivation and proofs but not in details. it seems correct to me

__ Clarity__: the paper is overall well-written. The conclusion or discussion about future work is missing. it is bit direct to finish the paper by an experimental sub-section.
Minor:
- in 175 UB_w and LB_w should be UB_q and LB_q ?

__ Relation to Prior Work__: The paper is closely related to MQL and MWL in Uhera et al 2019 and Algaedice et al in Nachum et al 2019.
I understand that there is page limit but I think the paper needs a background section to describe MQL and MWL and Algaedice algorithms. Especially, the paper uses different notations than Algaedice's paper, it is hard to parse the two papers to find out the connections between each methods.
In addition, I suspected that MUB-PO is the same as Frenchel Algaedice when the regularization coefficient \alpha in Algaedice is negative (maybe with \alpha=-1).
Could the authors comment on this please ?

__ Reproducibility__: Yes

__ Additional Feedback__: I am willing to increase my score if the authors address some of my comments.

__ Summary and Contributions__: #### Post author feedback ####
My main concern regarding the paper was not treating the sampling error. The authors convinced me in their response that it is a difficult problem and I don't think not having solved that should prevent acceptance of this paper. I have therefore increased my score to 7.
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The paper proposes a way to bound the off-policy evaluation errors when using one of the recently popular family of methods which estimate the stationary state distribution (or its importance sampling weights with respect to a dataset). The authors do this by deriving lower an upper bounds for estimation with respect to optimization of a function class approximating either weights or value function. The authors unify the two approximation methods by showing their equivalence as a min-max optimization problem.

__ Strengths__: The analysis method is fairly straightforward, but manages to give a strong intuition for understanding the uncertainty of the OPE estimators considered which stems from approximation difficulties and model misspecification. The unification of optimization in the space of weights/q-function is interesting, and so overall the paper makes a nice contribution to understanding the uncertainties of this class of OPE methods.

__ Weaknesses__: While the paper in its present form is already (marginally) good enough for publication, it still feels a bit preliminary and I would have preferred to see it published in a more mature version. I think the biggest contribution would come from considering the uncertainty introduced by sampling noise. The authors mention that they leave that for future work, but the paper would feel much more complete if this source of uncertainty would be considered, as it can be significant in practice.
The empirical results could be strengthened with more domains (and more complicated) rather than just cart-pole. While I think this would improve the paper, this is not a major weakness as there are still enough results to demonstrate the ideas presented in the paper.

__ Correctness__: The paper is correct to the extent I checked.

__ Clarity__: The paper is written clearly with minor typos which should be polished.

__ Relation to Prior Work__: The paper properly places itself in the context of prior work.

__ Reproducibility__: Yes

__ Additional Feedback__: This maybe an issue of taste, but I really don't like the abuse of the term "CI" for the bounds the authors are computing. I think it's misleading both in the sense that it does not consider the sampling noise, and doesn't have any statistical interpretation, and both of these elements are very characteristic of what I think most people usually think about when talking about CIs. i would have felt much more comfortable with a term like "approximation error bound" or something similar.

__ Summary and Contributions__: #### Post author feedback ####
The authors partially address my concerns in their rebuttal. I still think the experimental should be strengthed largely.
Therefore, I keep my score to 6.
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This paper mainly focuses on the off-policy evaluation for minimax methods that use value function and importance sampling. The authors develop the interval confidence interval under several circumstances. The interval confidence also connect two types of methods, weight learning and value learning. In the end, empirical results are used to show the validation of the proposed interval.

__ Strengths__: The paper investigates an important problem (off policy evalution) from the perspective of theortical confidence interval. As we see great success by using RL in games and real applications, theortical analysis of the performance and evalution is needed. Therefore, I personaly think this paper is a good try.

__ Weaknesses__: The emperiment scenerios are not rich to show the implication of the proposed confidence interval. More experiments can be added.

__ Correctness__: The logic is sound and supported by experiments.

__ Clarity__: The paper is easy to follow and clearly written.

__ Relation to Prior Work__: Related works are well addressed.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: This paper studies confidence interval (no uncertainty quantification but quantifies the bias due to the use of function approximations) for off-policy evaluation based on marginalized importance sampling. Due to the recent success of marginalized importance sampling for OPE problem, it is worth to understand deeper from a theoretical perspective.

__ Strengths__: I feel this paper provides us deeper understanding on function approximation for Q-function and marginalized importance weights. When either Q-function and marginalized importance weights are realizable in two function class, this paper provides a quantification on how the bias affects the estimation.

__ Weaknesses__: The study of bias issue is important, but I am not fully convinced the motivation of this so-called "confidence interval". Normally the confidence interval is designed for uncertain quantification and thus of great practical interest. However, although the authors explicitly point out they do not consider uncertainties, this will rule out all the important applications that typical CI could do (safe RL or else) (this CI will not be valid in practice due to estimation error). Thus, I can only view the contribution in this paper as sort of additional guarantee for the algorithm proposed in "Minimax Weight and Q-Function Learning for Off-Policy Evaluation" since the algorithms are the same. Solely quantifying a bias of an existing estimator may not be viewed as sufficiently significant. It is better to discuss after we identify those bias, can I reduce the bias in some way and improve existing algorithms?
The use of the derived CI for policy optimization is not very clear. Without considering the uncentaities from the sample, the CI is not vaild for practical use. Since this is batch setting, where does exploration/exploitation come from? This needs to be clarified since commonly exploration/exploitation refers to online setting.
The authors argue that the contributions over Liu et al. [5], Uehara et al. [1] are to make effective use of all components of data. This argument is not strongly supported in the following context (in theory or experiments).
Overall, there are many pieces of merits in this paper but may lack a strong point to support the contribution over Uehara et al. [1].

__ Correctness__: I think so.

__ Clarity__: Generally good. Section 5 may need some improvements.

__ Relation to Prior Work__: Good enough and clear.

__ Reproducibility__: No

__ Additional Feedback__: