Although I like the general idea of using DPP, I found there are various issues with the current version of the paper. Please see my detailed comments as follows.
• The paper specifically targets the permutation problems, but I don't see how this permutation property is incorporated into the design of the proposed acquisition function (except the fact that batch BO is used so that we can evaluate multiple data points parallel and thus can avoid the issue of large search space for permutation problems).
• Even though the paper provides various theoretical analysis, but these analyses seem not to be rigorous, and might not really help to answer the analytical properties of the proposed approach.
o The Acquisition Weighted Kernel L^{AW} defined in Line 122 seems to not be a real (valid) kernel? As it depends on any acquisition function a(x), so it seems impossible to me that it is a valid kernel for all cases. Besides, this seems to be like a component of the proposed acquisition function, rather than to be called a "kernel".
o The regret analysis in Theorem 3.6 depends on the maximum information gain \gamma_T, but this \gamma_T value is not properly bounded in Theorem 3.9. Theorem 3.9 only shows that \gamma_T is smaller than a function of \lambda_{max} but there is no guarantee that \lambda_{max} is bounded when T goes to infinity. This is a key analysis in any BO analysis. In the literature, only several kernels have been shown that their \lambda_{max} is bounded when T goes to infinity.
o Again, same problem with Theorem 3.12, is there any guarantee that the \lambda_{max} of the position kernel is upper bounded?
• The performance of the proposed approach on the permutation optimization problems are not that good though (Section 5.1). LAW-EI performs pretty bad, much worse than other baselines in various cases, while LAW-EST is on par with other baselines and only performs well in one problem. Besides, I don't understand why LAW-UCB is not added as one of the baselines. The justification regarding the size of search space does not seem reasonable to me.
• Besides, the experiments seem not too strong and fair to me. I don't understand why all the baselines use the position kernels, why don't we use the default settings of these baselines in the literature? Besides, it seems like some baselines related to BO with discrete & categorial variables are missing. The paper also needs to compare its proposed approach with these baselines.
I think the paper does not mention much about the limitations or the societal impacts of their proposed approach.
• Besides, the experiments seem not too strong and fair to me. I don't understand why all the baselines use the position kernels, why don't we use the default settings of these baselines in the literature? Besides, it seems like some baselines related to BO with discrete & categorial variables are missing. The paper also needs to compare its proposed approach with these baselines. I think the paper does not mention much about the limitations or the societal impacts of their proposed approach.