The authors provide a clear descriptions of their setting and methods. They clearly point out where they can make improvements and how this setting is novel. They introduce a new structured bandit problem that is interesting and could have a significant impact on the field. Line 163, it would be more easy to read if you change \Delta_{\hat{m}, i} to \Delta_{\hat{m}, j} as this would reflect the naming structure from your previous part of the paper.

This paper proposes a novel problem setting about the model specification in the multi-armed bandit problem. The method proposed by authors are quite persuasive, and the theoretical result consistent with the expected theoretical ideal result predicted from previous researches of the linear bandit. However, I have two concerns. First, I could not understand why authors dervise their algorithm from a method proposed in the research of adversarial bandit problem. Second, it is not necessary, but authors are recommended to show their experimental results. I think that it is not so difficult to experiment with their problem setting. Originality: 5

Overall, this paper presents a novel algorithm for a challenging problem with solid theoretical analysis. The theorems are well supported with theoretical proofs. The results are promising. The paper is well written and clearly organized. All contributions are well presented and explained in the paper. And I like the discussion provided in this about the intuition behind the algorithm and some remaining questions. The discussion helps readers to understand the main idea better.