Paper:
Published as a conference paper at ICLR 2018 DEEP BAYESIAN BANDITS SHOWDOWN AN EMPIRICAL COMPARISON OF BAYESIAN DEEP NETWORKS FOR THOMPSON SAMPLING
Recent advances in deep reinforcement learning have made significant strides in performance on applications such as Go and Atari games. However, developing practical methods to balance exploration and exploitation in complex domains remains largely unsolved. Thompson Sampling and its extension to reinforcement learning provide an elegant approach to exploration that only requires access to posterior samples of the model. At the same time, advances in approximate Bayesian methods have made posterior approximation for flexible neural network models practical. Thus, it is attractive to consider approximate Bayesian neural networks in a Thompson Sampling framework. To understand the impact of using an approximate posterior on Thompson Sampling, we benchmark well-established and recently developed methods for approximate posterior sampling combined with Thompson Sampling over a series of contextual bandit problems. We found that many approaches that have been successful in the supervised learning setting underperformed in the sequential decision-making scenario. In particular, we highlight the challenge of adapting slowly converging uncertainty estimates to the online setting.

Review:
If two major questions below are answered affirmatively, I believe this article could be very good contribution to the field and deserve publication in ICLR.
In this article the authors provide a service to the community by comparing the current most used algorithms for Thompson Sampling-based contextual (parametric) bandits on clear empirical benchmark. They reimplement the key algorithms, investing time to make up for the lack of published source code for some. 
After a clear exposure of the reasons why Thompson Sampling is attractive, they overview concisely the key ideas behind 7 different families of algorithms, with proper literature review. They highlight some of the subtleties of benchmarking bandit problems (or any active learning algorithms for that matter): the lack of counterfactual and hence the difference in observed datasets. They explain their benchmark framework and datasets, then briefly summarise the results for each class of algorithms. Most of the actual measures from the benchmark are provided in a lengthy appendix 12 pages appendix choke-full of graphs and tables.
It is refreshing to see an article that does not boast to offer the new "bestest-ever" algorithm in town, overcrowding a landscape, but instead tries to prune the tree of possibilities and wading through other people's inflated claims. To the authors: thank you! It is too easy to dismiss these articles as "pedestrian non-innovative groundwork": if there were more like it, our field would certainly be more readable and less novelty-prone.
Of course, there is no perfect benchmark, and like every benchmark, the choices made by the authors could be debated to no end. At least, the authors try to explain them, and the tradeoffs they faced, as clearly as possible (except for two points mentioned below), which again is too rare in our field. 
Major clarifications needed:
My two key questions are:
* Is the code of good quality, with exact  reproducibility and good potential extension in a standard language (e.g. Python)? This benchmark only gets its full interest if the code is publicised and well engineered. The open-sourcing is planned, according to footnote 1, is planned -- but this should be made clearer in the main text. There is no discussion of the engineering quality, not even of the language used, and this is quite important if the authors want the community to build upon this work. The code was not submitted for review, and as such its accessibility to new contributors is unknown to this reviewer. That could be a make or break feature of this work. 
* Is the hyper parameter tuning reproducible? Hyperparameter tuning should be discussed much more clearly (in the Appendix): while I appreciate the discussion page 8 of how they were frozen across datasets, "they were chosen through careful tuning" is way too short. What kind of tuning? Was it  manual, and hence not reproducible? Or was it a clear, reproducible grid search or optimiser? I thoroughly hope for the later, otherwise an unreproducible benchmark would be very 
If the answers to the two questions above is "YES", then brilliant article, I am ready to increase my score. However, if either is a "NO", I am afraid that would limit to how much this benchmark will serve as a reference (as opposed to "just one interesting datapoint").
Minor improvements:
* Please proofread some obvious typos: 
  - page 4 "suggesed" -> "suggested",  
  - page 8 runaway math environment wreaking the end of the sentence.
  - reference "Meire Fortunato (2017)" should be  "Fortunato et al. (2017)", throughout.
* Improve readability of figures' legends, e.g. Figure 2.(b) key is un-readable. 
* A simple table mapping the name of the algorithm to the corresponding article is missing. Not everyone knows what BBB and BBBN stands for.
* A measure of wall time would be needed: while computational cost is often mentioned (especially as a drawback to getting proper performance out of variational inference), it is nowhere plotted. Of course that would partly depend on the quality of the implementation, but this is somewhat mitigated if all the algorithms have been reimplemented by the authors (is that the case? please clarify).