__ Summary and Contributions__: This paper studies Linear Contextual Bandit problem with context and set of available actions, generated by oblivious adversary, and a fixed loss functions. This is the first work, that considers infinite action set for this setting. The main focus of this work is on the misspecified case with unknown upper bound on the misspecification, that wasn’t studied before. The solution of this problem is based on the corralling procedure. Other contribution of this work is replacing dimension d in epsilon \sqrt{d} T term in the regret to the effective dimension of the sequence of sets of available actions.

__ Strengths__: I find presented corralling procedure potentially useful for future works, like model selection.
The result on the replacing dimension with effective dimension gives a big improvement for sparse problems.

__ Weaknesses__: Optimisation problem in Definition 7 doesn’t look computationally efficient. Could you add more details on how do you propose to solve this problem?

__ Correctness__: The analysis looks correct to me.

__ Clarity__: Some problems with numeration of theorems, assumptions, etc. Apart from that the paper is well written.

__ Relation to Prior Work__: Foster and Rakhlin, 2020 shows epsilon\sqrt{K}T term in the misspecification, which can be better than effective demention in many cases. Is it possible to get min(K, d) in your work?

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: This paper studies the misspecification issue for linear contextual bandits with a regression oracle. Specifically, built on [21], the authors extend the algorithm and the analysis to address the challenges of infinite action sets and the unknown misspecification level of the function class.
The main contributions of this paper are three-fold: (i) It proposes to use the logdet-barrier distribution to address the infinite action sets; (ii) A new master algorithm is proposed to combine base bandit algorithms of different misspecification levels; (iii) It presents rigorous regret analysis of the proposed algorithms.
===== Post-Rebuttal =====
The authors' response addressed the major concerns. I have updated the score accordingly.

__ Strengths__: 1. The bandit setting considered in this paper is important.
2. The proposed base algorithm ImpSquareCB and the master algorithm (\alpha, R)-hedged FTRL are novel.
3. The regret analysis is rigorous and non-trivial.

__ Weaknesses__: 1. The presentation can be improved:
- Several designs are introduced without much intuition. For example, in Algorithm 2, why is logdet-barrier a proper distribution for selecting the actions? What’s the intuition behind the design of the learning rate $\gamma_t$?
- There is a big leap from Section 3.2 to Section 3.3. First, it is not well explained why the concept of importance-weighted regret is needed or why a master algorithm is required to address unknown misspecification. Section 3.3 is presented in a bottom-up manner and turns out to be a bit confusing.
- In Lines 166-167, it is mentioned that adapting SquareCB to the misspecification setting is non-trivial, but without any explanation.
2. The lack of empirical results:
While I usually do not comment on the lack of empirical results, for this topic a few sets of empirical results (e.g. synthetic experiments) will be very helpful in demonstrating the performance gain of using the proposed method, compared to other benchmarks that require either a realizability assumption or the knowledge of misspecification level.

__ Correctness__: I have read through the proofs up to Section 4 and had a quick look at the proofs in Sections 5. While I do not find any specific errors in the proofs, there are indeed several equations that can be made more transparent (please see detailed comments below).

__ Clarity__: Overall this paper is clearly written, except for Section 3.3 as pointed out above.

__ Relation to Prior Work__: This paper has correctly cited the prior works and included all the important references.

__ Reproducibility__: Yes

__ Additional Feedback__: Some additional comments:
Lines 164-165: The first claim in Remark 9 is presented without proof. This could be made more transparent at least by providing a proof sketch.
Lines 183-184: Theorem 10 seems to presume that the function class consists of constant vector-valued functions based on the proof in Appendix C (Lines 486-487). However, from the description of Theorem 10, it is unclear what function class is considered.
Line 205: This equation of regret is claimed but with its proof postponed to Line 500. It would be helpful to mention this right after Line 205.
Line 451: How to obtain the last two equalities? In the second last equality, why is the expectation independent of $\hat{\theta}$? Please make this more transparent.
Algorithms 1 and 2: The term “SqAlg” is not defined.
Line 487: Regarding Reg_{Imp}, how to obtain the second equation from the first one?

__ Summary and Contributions__: This paper tackles the \varepsilon misspecified bandits without prior knowledge about misspecification level for infinite arms. With a regression oracle, optimal regret bound is proved. Moreover, they have derived a class of improved master algorithms for corralling. The reviewer likes the discussion about connection with adversarial corrupted bandits.

__ Strengths__: The proofs and claims are sound. Although the topic related to misspecified bandit is not a new problem and has been well investigated, this paper is still worthwhile in some perspectives, e.g., the open problems from [27] with unknown \varepsilon and the improved master algorithms for corralling. This paper is relevance to the NeurIPS community.

__ Weaknesses__: (1) The regret bound is still linear with T unless more assumption about \epsilon. Does this mean in practice misspecification will heavily harm the performance (always negative result)? This is a little different from other misspecified bandits. Could the authors give a non-regret setting ? e.g., the small deviation case of [22].

__ Correctness__: The claims and method are correct.

__ Clarity__: This paper is well written.

__ Relation to Prior Work__: This paper clearly discussed how this work differs from previous contributions.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The paper considers the misspecified linear contextual bandit problem and proposes a novel contextual bandit algorithm robust to misspecification. The algorithm relies on calls to an optimal online regression oracle, and when specialized to linear contextual bandits with infinite actions achieves optimal regret guarantees that decompose additively into two terms, one of them depending linearly on the misspecification parameter. Moreover, the algorithm does not need to know the true misspecification parameter but can adapt to misspecification.

__ Strengths__: This paper solves an interesting open problem posed in the work of Lattimore et al., for which the previous phased elimination type algorithms do not easily extend. The idea of this work is to consider another approach inspired by Foster and Rakhlin, and their idea of the reduction of contextual bandit to online regression. One of the main ideas in this paper of replacing abe-long with log-barrier (which allows them to handle infinite action sets) seems novel. Further on, non-trivial work is required to obtain the regret bound that matches the existing lower one. In particular, a novel algorithm based on Corraling + weighted modification of SquareCB is proposed. Finally, model mismatch in bandits and RL is a very important problem that has recently received a lot of attention. Hence, I think that this paper meets the requirements when it comes to both novelty and significance.

__ Weaknesses__: I did not spot any major limitations/weaknesses in this work. However, the authors state that the algorithm is suitable for practical deployment, and although this a theoretical work, simulations/experiments that support this claim would be a plus. Also related to this, the overall computational complexity of the proposed method seems not discussed.

__ Correctness__: From a limited inspection, the claims seem to be correct. The statement of Theorem 12 contains an additional sqrt{d} in the first term. By inspecting its proof, this seems to be a typo.

__ Clarity__: Overall, the paper is well-written. Some suggestions for improvement are outlined in the "additional feedback" section.

__ Relation to Prior Work__: I think that the authors have clearly explained the previous work and differences concerning the previous works of Foster and Rakhlin and SquareCB, and the work of Agarwal et al. They have also used quite some space in the main body of the paper to explain these methods which definitely improves readability. The difference with respect to the work of Lattimore et al. is clearly explained and some further related work on the adversarially corrupted bandits is provided.

__ Reproducibility__: Yes

__ Additional Feedback__: ________________________________
Post-rebuttal comments: After reading the rebuttal and other reviews, I keep my score. The main theoretical result is both novel and strong. There are some good suggestions when it comes to improving the overall clarity of the paper in the reviews and I hope the authors would consider them in the revised version.
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When explaining Algorithm 3, it is not clear how q_t gets updated (also in the provided pseudocode). This is only discussed later on in the paper. Perhaps, provide a reference to the section where this is discussed. Also, the paper would further benefit from more intution/discussion on the need for weighted regression oracle.
Can you also comment on the computational complexity of the overall method? Can you explain the sampling procedure in Alg.2, i.e., a_t ~logdet-barrier, and the need of solving the problem in Definition 7 (i.e., its solution and method that solves it)?
I’m curious about Assumption 1 in other settings rather than linear, e.g., the mentioned kernelized setting? Can you perhaps comment on the extension of your result to this setting? Does it make sense to talk about misspecification setting in the case of universal kernels such as, e.g., Gaussian kernel?
Minor comments/typos:
- Eq. 2, perhaps A_t instead of A(x_t)
- 135, comma is missing {e1, …, ek}
- Alg. 1, gamma is not defined/mentioned earlier
- Eq. 3, both i^* and a^* are used.
- 162, 2 x “the”
- SqAlg is not defined
- 172, here you talk about Bernoulli and q_t is its parameter, later on (e.g., 197) A_t ~q_t.
- Algorithm 3, rho_t,A_t also needs to be passed
- 201, empty space