__ Summary and Contributions__: This paper studies the stochastic partial monitoring (PM) problem using Thompson sampling (TS). The author designs a new TS-based algorithm that can sample from the exact posterior distribution. On the theoretical side, the paper provides the first logarithmic regret bound for a TS-based algorithm for linear bandit problems and strongly locally observable PM games. Numerically, simulations show that the proposed algorithm outperforms the previous methods.

__ Strengths__: The paper provides the first theoretical guarantee that achieves logarithmic regret bound using TS-based sampling. Furthermore, the author performs experiments and show the advantage of the new algorithm over the existing methods.

__ Weaknesses__: This main weakness of this paper is that it only focuses on the finite stochastic PM games while other general settings such as sub-Gaussian is of great importance.

__ Correctness__: The technical part of this paper looks correct to me. The experimental part also looks solid. However, I didn't get a chance to verify the details in the supplementary materials.

__ Clarity__: The paper is nicely written with a comprehensive overview of the problem and clearly presented technical ingredients.

__ Relation to Prior Work__: The paper clearly addresses related works on PM and other methods of sampling from the posterior distributions. There are only a few previous works on analyzing TS and PM due to the intrinsic difficulty of the problem.

__ Reproducibility__: Yes

__ Additional Feedback__: After the response phase, considering the additional feedback received, I remain with my initial assessment of the paper.

__ Summary and Contributions__: This paper considers the application of Thompson Sampling (TS) to stochastic finite partial monitoring (PM). The authors propose a TS scheme using accept-reject sampling, with a parameter which can be tuned to trade-off the accuracy of the sampler for improved computational efficiency. The authors then introduce a "linearized" variant of the finite PM problem where the observed signals have additional (unit variance) Gaussian noise. One feature of this variant is that the accept-reject routine is not required as the posterior matches the proposal of the accept-reject scheme. The authors derive an instance-dependent O(log(T)) bound on the regret of their TS approach for this linearized problem (not the standard finite partial monitoring framework). In experiments, the proposed TS compares favourably to existing stochastic PM algorithms of Piccoloboni and Schindlehauer (2001) and Vanchinathan et al. (2014).

__ Strengths__: The paper gives an interesting and new result. The understanding of TS for PM is, as the authors point out, presently limited and this analysis improves our understanding of the extent to which it is effective. This insight is relevant to the NeurIPS community. While I have not examined the entire supplement in rigorous detail, it seems that the proof uses interesting ideas, nicely exposited on lines 251-262. Further the experiments display that the method considered substantially improves over the competitors that were investigated.

__ Weaknesses__: I think the makings of a very good paper are here, but there are several ways in which it could be improved.
Firstly, there is the issue of the mismatch between the setting of the theoretical results and the experimental section/broader focus of the paper. The algorithm is initially posed in a different setting (called "discrete setting") to the theoretical results, and then the experimental section also, as I understand, considers the discrete setting. Why not make the entire focus of the paper the setting where theoretical results can be obtained?
Further, around this point, it could be clearer why results cannot be obtained in the discrete setting, and why the noise in the linearized setting needs to have unit variance as described on line 200. If this is the case the scope of the theoretical contribution seems to be somewhat limited.
Second, I believe that the Mario Sampling algorithm of Lattimore and Szepesvari (2019) "An Information-Theoretic Approach to Minimax Regret in Partial Monitoring" would be applicable in this setting. The experimental section therefore suffers from incompleteness by not acknowledging this approach. A more complete picture of the performance of TS relative to other approaches would be provided by also comparing to this scheme.
Finally, I feel that while the results are themselves undoubtedly interesting to the NeurIPS community, the size of the contribution is perhaps a bit smaller than the average good NeurIPS paper. As said previously the setting in which the results hold feels somewhat limited. If results for the discrete setting could be established, or further insight as to either why this is challenging were reported, for instance, this would represent a more substantial contribution.

__ Correctness__: I was not able to find issues with the correctness of the work.

__ Clarity__: Yes, the paper is mostly well written. The authors display a clear knowledge of theoretical analysis of partial monitoring and cover the key concepts. There are areas where the writing could be more purposeful. For instance in lines 43-45, some papers containing algorithms are listed but there is no real insight as to what these algorithms entail. Similarly, in lines 130-139 a number of sampling procedures are mentioned, but without any real discussion of their relative pros or cons.

__ Relation to Prior Work__: As I mention in the weaknesses section, I think there is a missing comparison to the PM algorithm of Lattimore and Szepesvari. Less critically, I think the discussion of existing results around TS is missing a line of work following from Russo and Van Roy's information-theoretic analysis of the Bayesian regret of TS. A fuller picture of existing results on the performance of TS could be given by reviewing this line of work.

__ Reproducibility__: Yes

__ Additional Feedback__: UPDATE: Thank you for your response and effectively addressing my concerns. In light of the rebuttal, other reviews and some reflection I have increased my score. I think that there is a sufficient amount of interesting content to merit publication.
Some minor comments:
In the comments on the upper bound it would be useful to have some sense of the magnitude of the z_{j,k} terms.
In line 216 it is unclear whether the algorithm is different from the family of TS used in practical settings because of the aforementioned dependence on the time horizon, or some other reason.
Lines 244-250 are not the clearest to the reader. Eventually it becomes clear that this is a discussion of the mechanisms used in the proof, but this could be established more clearly.

__ Summary and Contributions__: The subject of the paper is finite stochastic partial monitoring (PM): at each stage an agent chooses an action in a finite set, an opponent does likewise in another finite set (according to an unknown probability distribution) and the agent gets to observe a signal depending on both actions. the agent also incurs an unobserved loss depending on these same actions.
The setting of discrete PM is introduced (where the number of signals is finite) and a thompson-sampling based algorithm is presented. That algorithm uses an accept-reject procedure to sample from the posterior by using samples from a Gaussian auxiliary distribution.
The algorithm is analysed in a different setting, not discrete PM but linear PM, in which the signals are not discrete but have a linear structure. A O(log T) expected regret is shown. As a particular case, this is the first log(T) regret bound for Thompson sampling for linear bandits.

__ Strengths__: The algorithm is well described, uses an interesting method to sample exactly from the posterior, and the authors obtain the first regret bounds for a Thompson sampling approach in that setting.
The empirical performance of the algorithm on the discrete PM problem is much better than existing approaches.
The first logarithmic bound for Thompson sampling for linear bandits is a significant contribution to the bandit literature.

__ Weaknesses__: The bound is only valid for the linear PM problem with Gaussian noise, and not for sub-Gaussian noise as is often the case with bandit algorithms (which would also include the discrete PM case).
From the experimental evaluation, it looks like the rejection sampling procedures suffers from many rejects when the number of actions gets larger, such that the approach becomes less practical. This is not discussed in the paper.

__ Correctness__: I did not read the entirety of the analysis in the appendix but only checked a few points. What I checked was correct.

__ Clarity__: The introduction of the setting of discrete PM and the description of the algorithm are very clear. The experiments are well presented as well.
The setting changes suddenly at the bottom of page 5 from discrete partial monitoring to Gaussian linear. The transition is abrupt and not well hinted at before. Indeed there is even the misleading statement of line 145: "we also give theoretical analysis for the proposed algorithm". This is not true in the discrete setting under discussion at that point.
A detail: in definition 3, k is a symbol, while in definition 4 it is an action. Clarity would be improved by associating more strongly notations and concepts.

__ Relation to Prior Work__: Prior work is clearly discussed.

__ Reproducibility__: Yes

__ Additional Feedback__: Line 287: "Note that our algorithm can also be applied to a hard game, though there is no theoretical guarantee". Since the problems used in the experimental section are of the discrete type, there is also no guarantee for the easy case.

__ Summary and Contributions__: The paper studies stochastic bandits with a finite set of actions. It makes two novelty claims:
1. An argument that Thompson sampling achieves gap-dependent bound of sum(Delta_i / Delta_min^2 * log(T)).
2. An approximation algorithm of Thompson sampling posteriors with speed-performance trade-offs.
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After reading the other reviews and author feedback, I feel more associated with the motivations in this work. I want to see two points of improvements before camera ready:
1. The introduction is lacking concrete examples. What is partial monitoring? Why is it important? What applications would this lead to? Research contributions should be measured by the amount of new opportunities it may generate. To give readers room for imagination, one strategy would be to provide numerical examples right after each motivation claim.
2. The sampling strategy seems to have some limitations due to curse of dimensionality. I would like to see a discussion of the limitations and why that does not matter in actual application of this work.
Good writing should make their work motivating, accessible, and enjoyable. Even in pure math, people still find direct applications, e.g., in cryptography, to make their impacts obvious. In the latter, people may make a bigger effort to appreciate the work because they feel associated.

__ Strengths__: Relevance to the NeurIPS community: Bandits algorithm and analysis is of interest to the community.
Clarity: The paper has a clear organization.

__ Weaknesses__: Novelty: The claim #1 is expected from very early results. The claim #2 is unclear because the number of rejection times seems high, which indicates that the approximation algorithm may be inefficient.
Empirical: The paper includes experiments, but they are limited in scale. The lack of contextual feature modeling also makes the work difficult to apply in practice.

__ Correctness__: I did not read the appendix but the result agrees with my expectations based on related work.

__ Clarity__: Yes

__ Relation to Prior Work__: No. I would expect to see three points of clarifications:
1. The main theoretical novelty came from a claim that a high-probably regret bound cannot yield a regret bound in expectation (line 223). This is surprising.
2. The complaint against "a time-consuming optimization problem" needs to be elaborated. Particularly, how does it relate to the objective function (1)?
3. The discussion about Bartok et al., 2011 is unspecific. The author aims to classify minimax regrets to trivial, easy, hard, and hopeless, but did not give any intuitions.

__ Reproducibility__: Yes

__ Additional Feedback__: * I am not seeing the point to mention PM games. All bandits work can be framed as minimax games. How is this work different? This confused me a little in the introduction.
* Definition 1 through Definition 5 could be mostly deleted without impairing understanding. What is a Pareto front doing here?
* Let me know if I understood the contributions correctly. If so, Section 3.1 needs to be clarified. For example, the rejection function is not presented.
* Section 3.2 seems intuitively useful, but it is completely out of context. I am not clear if it makes a difference with or without rank reductions.
* Experiments setup. Needs to remind reader what is N and M. (I guess number of arms?) How can they be different?
* There is a general lack of discussions of the results - how different conditions impact the cumulative regrets. The image backgrounds also should be set as white or transparent.
* Could you add the theoretical regret curve in this experiment for comparisons? If it incurs large constants, then you could draw it in a twinx plot. I am primarily hoping to see the logarithmic rates.