__ Summary and Contributions__: High-dimensional optimization problems are still challenging for Bayesian optimization methods. Linear embeddings are one alternative to handle this setting, but suffer from several limitations: (i) the embedding might produce representations that are hard to model with a Gaussian Process, (ii) points chosen in the embedding map outside the bounds of the optimization problem, (iii) product kernels are not preserved in the embedded space and (iv) the embedding might not contain the optimum. This paper studies these limitations and proposes a solution, called ALEBO, (i) using a different kernel, (ii) searching only that portion of the embedded space that can be mapped to the original bounds and (iii) maintaining a high probability that the optimum lies in the embedding. The paper compares ALEBO to existing high-dimensional BO methods, such as REMBO, SMAC, TurBO and HeSBO but also other global optimization methods such as CMA-ES.

__ Strengths__: High-dimensional optimization with a high sample efficiency, e.g. for tuning hyperparameters of machine learning models, is still challenging and a highly relevant topic for the NeurIPS community. REMBO and HeSBO make use of embeddings and thus approach high-dimensional problems. However, these methods still suffer some limitations. This paper explains and studies what causes these limitations, how they affect practical applications and propose a solution that overcomes these problems.

__ Weaknesses__: The paper proposed a convincing method, but I have two criticisms: While the paper proposed a thorough analysis and gives detailed explanations of limitations of prior work, the overall contribution is rather incremental and does not yield a highly competitive baseline compared to TurBO and CMA-ES: On the constrained NAS problem not all methods were evaluated (see my comments below) and on the policy search benchmark another BO method, TuRBO, seems to perform better. Furthermore, there also remains the open question of how to select the dimensionality of the embedding in practice, which potentially dramatically impacts the optimization performance.

__ Correctness__: The claims and method seem correct to me.

__ Clarity__: The paper is overall well written, however, parts of the supplemental are critical to understand the paper. I am aware of the space restrictions, but I would recommend to move Algorithm 1 to the main paper since I found it very helpful to understand how the different parts of the method work together. Furthermore, I would also replace Figure 7 (right) with Figure S11 from the supplemental since it shows the variance across repetitions.

__ Relation to Prior Work__: To the best of my knowledge prior work is properly addressed and discussed.

__ Reproducibility__: Yes

__ Additional Feedback__: Q1: Does the NASBench 101 benchmark require using the constraint that the training time is less than 30min? I agree, that constrained optimization is an important topic in BO, but in my opinion, this is not the main focus of this paper. This design decision does not allow to compare to other optimizers and so I am wondering how these would compare on this benchmark if there are no constraints?
Q2: How would one set d_e in practice. Do you have any insights whether ALEBO is more or less sensitive wrt. to this hyperparameter compared to prior work?
## After readings the author's response
I'd like to thank the authors for addressing my questions and concerns. I appreciate adding a discussion about the flexibility of the proposed method and extending the set of optimization methods compared on the the real-world problems, which, I believe, will be a great addition and significantly strengthens the contribution.

__ Summary and Contributions__: In this article the authors revisit the use of random linear embeddings in Bayesian optimization. They identify key aspects explaining some mitigated results in benchmarks. Improvements include restriction to polytope bounds and the use of an appropriate kernel to model projected observations (and handle distortion). Benchmark results on synthetic and realistic problems show promising results.

__ Strengths__: I found the paper to be well structured and pedagogical given the page limit. The performance of the proposed method is good over several tests and include relevant competitors.

__ Weaknesses__: It could be worth detailing more the limitations: arbitrary selection of de, longer training time with Mahalanobis kernel, limits of linear embeddings.

__ Correctness__: The paper comes with backing simulation results and extensive tests.

__ Clarity__: I would move the algorithmic summary in the paper to fix some ideas (e.g., that B is fixed after sampling once).

__ Relation to Prior Work__: A good summary of existing related works is provided.

__ Reproducibility__: Yes

__ Additional Feedback__: Some statements could be clarified:
L95: is k=4 always used?
L213: it does for axis-aligned embeddings
And I have several questions and suggestions for improvements:
a) How to choose de in practice, given that the number of hyperparameters grows quadratically? In the example around line 230, retaining 12 to 20 variables is necessary with a relatively optimistic prior on the location of the optimum (it is not uncommon that several influential linear combinations of variables reach their bounds), but this is already quite high-dimensional for GP modeling.
c) For testing the proposed Mahalanobis kernel, rather than testing on the Hartmann 6 function were most variables are inactive, it would be more pertinent to work on a sample from a D-dimensional ARD kernel. Tests could include rotation with just d-active ARD variables (with or without rotation) but also a non-zero activity for the D-d ones to be more realistic.
### Additions after rebuttal ###
I thank the authors for addressing my questions. I am also appreciate the propositions for the additional page with more results and discussion. Based on the analysis of pitfalls of random embeddings, this could help defining a better baseline for new high-dim BO methods compared to the naive implementation of REMBO.

__ Summary and Contributions__: This paper closely looks into linear embeddings to scale up Bayesian optimization (BO) to high dimensions. The authors give a detailed analysis of why existing linear embedding techniques may fall short, and builds on these insights to propose ALEBO. This new linear embedding approach relies on a Mahalanobis kernel in tandem with a hypersphere sampling to increase the probability that the optimal solution exists in the embedding. Experiments on synthetic data as well as on two real-world scenarios show that ALEBO effectively fixes many of the shortcomings of linear embeddings and improves their performance.

__ Strengths__: 1. This is the first paper to perform an in-depth analysis of several shortcomings of linear embeddings through an extensive empirical evaluation and ablation studies. I expect these insights to benefit the AutoML community and steer future work on high-dimensional BO in the right directions (potentially away from linear embeddings).
2. The paper introduces ALEBO, which alleviates the limitations of linear embedding approaches and is a step forward making them competitive with the state of the art.

__ Weaknesses__: 1. Despite the proposed improvements, the real-world experiments show that linear embeddings still do not emerge as a competitive approach in practice. To some extent, this narrows down the scope and impact of the contribution.

__ Correctness__: The claims are correct and all empirically grounded by ablation studies and comparisons with baselines.

__ Clarity__: The presentation is excellent: well organized, crisp and clear. The paper was a pleasure to read.

__ Relation to Prior Work__: Prior work is carefully discussed and an extensive set of baselines compared against. The experiments do miss some baselines from the family of methods making use of surrogate models to scale up to high dimensions. An example is Rolland et al., 2018, which would have been a strictly better and more interesting baseline compared to Add-GP-UBC.

__ Reproducibility__: Yes

__ Additional Feedback__: I think this is a good paper that will inform future work on high-dimensional BO. Having highlighted a number of severe shortcomings of linear embeddings, I expect future work to either leverage the insights of ALEBO to develop a truly competitive baseline, or simply use these lessons learned to focus on different methods, such as the model-free ones.
The robot locomotion experiment does suggest that linear embeddings, despite all improvements, are still not suited to be the default for high dimensional BO. Not only are they outperformed by model-free methods, such as CMA-ES, but also by some model-based ones such as TuRBO (despite the larger variance, as shown in the appendix). In any case, while we do not have a new state of the art method for high-dimensional BO out of this paper, the contribution is useful and will inform future work in this space.
Some questions are below.
1. Figure 6: only a subset of methods are benchmarked here compared to Figure 7. I assume the reason is that only methods that support constrained optimization are considered. Have the authors also tried real-world experiments where the stronger baselines, such as CMA-ES, can actually be applied? How does ALEBO compare to them in these settings? Is it still outperformed as in Figure 7?
2. As Figure 6 focuses on a constrained optimization problem, how would approaches based on simple BO with constrained acquisition functions perform? Some examples include constrained expected improvement, constrained predictive entropy search, and constrained max-value entropy search (Gardner et al., 2014; Lobato et al., 2015; Perrone et al., 2019).
3. The authors focused the discussion on linear embedding methods, leaving nonlinear methods out of the scope of the work. Still, I would have liked to see more discussion on nonlinear methods based on VAEs. How do these compare to ALEBO? What learnings about the shortcomings of linear embeddings transfer to the nonlinear case?
4. Table S1 in the appendix shows that ALEBO's runtime scales very well in the problem dimension, but otherwise does not compare particularly well with several baselines. I suppose this is due to the complications implied by the linear constraints and the Mahalanobis kernel, which requires MCMC sampling and moment matching. How would ALEBO perform when the x axis is wall-clock time rather than iteration count? CMA-ES and Sobol would be easily winners when blackbox evaluations incur reasonable runtime, but then also REMBO and HeSBO could be more competitive considering they can take up to 40 times less compute time per iteration.
*** Post-rebuttal ***
I'd like to thank the authors for addressing my questions. I also appreciate that there will be an additional discussion around the connection with non-linear embeddings, which I believe will further increase the impact of this work. I would like to see this paper accepted and confirm my score.

__ Summary and Contributions__: The paper explains an approach for embedding high dimensional bayesian optimization problems that circunvents several issues found before.

__ Strengths__: Explainability
Clarity
Significance

__ Weaknesses__: Novelty
Theoretical work

__ Correctness__: Yes I think

__ Clarity__: Yes, it is excellently written, it a different style that is very didactic and easy to follow.

__ Relation to Prior Work__: Yes, and it examines several issues found before proposing solutions.

__ Reproducibility__: Yes

__ Additional Feedback__: Author rebuttal: I have read the author's response and talk with the other reviewers about this paper.
I keep my score into 7, I think this is a good paper.
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Good paper exposing several issues found in High BO that is given along with its code to ensure reproducibility. I have found your work very interesting. Although, I have some doubts that must be answered if more grade wants to be achieven.
For future work I suggest the extension of this work to multi-objective BO.
Concerning the constraints introduced by Eq 1, would the optimum still be accesible in that regions? Can it be proven theoretically? I found this aspect the most critical thing of the methodology presented in this paper. I have read the probability section of finding the optimum but I still miss a clear comparative of this probability wrt previous approaches and if, in practice, is high or low.
Is proposition 1 valid for any stationary kernel or it this entire result constrained to the ARD kernel substituting the euclidean distance for a Mahalanobis one? It would be great to generalize to any kernel to give the method more flexibility.
Why is TuRBO not tested in the CIFAR10 experiment?