Venue: CoRL 2022 Conference

Reviews:

final_score: rating: Weak Accept: I recommend accepting the paper, but will not argue for my recommendation if the majority of other reviewers have a different opinion.
review_content: summary: This paper proposes a method for synthesizing control barrier functions (CBF) that avoids action saturation for safe controllers. The main contribution is the proposal of a limit-aware CBF, and the usage of a min-max learning framework that uses a critic to find worst-case saturation points and a learner for updating the CBF. They also propose two neural CBF designs, an efficient method for optimizing the learner as a discrete minimizing problem, and introduce a regularization term to encourage larger safety sets. The paper compares their method to a non-neural CBF baseline for a pendulum-balancing task on a quadcopter, and found that their method enables nearly 100% of the boundary states of the safe set to avoid input saturation and nearly 100% of the rollouts are forward invariant inside the safe set, which is a significant improvement over the baseline.
strengths_and_weaknesses: Strengths
* The introduction of a non-saturating CBF to avoid input-saturation for safe controllers, which is an important consideration when synthesizing CBFs.
* The two proposed neural CBF designs are mathematically well motivated by the three constraints they need to satisfy to be valid safety specifications.
* The two strategies of taking the best of a batch and warmstarting the optimizing process for finding the states along the safe set boundary where saturation occurs is a useful inclusion to the process for improving the critic.
* The re-expression of the saturation risk as an analytical function and framing the learner update rule as a discrete minimization problem over counterexamples is both efficient and well-thought out.
* The regularization term that encourages states near the safe set boundary to move inside the safe set (or further in) is clever and well-designed in this paper.

Weaknesses:
* This paper does not implement their method on robot hardware, which makes it hard to determine how effective it is in practice due to the limiting assumptions on known and deterministic dynamics. In lieu of this, it would be worthwhile and more convincing to at least see simulated experiments with noise added to the dynamics to see how the learned neural CBF performs. 

robotics_focus: Highly relevant to robotics but no hardware experiments
quality_of_the_limitations_section: Limitations are addressed clearly
summary_of_recommendation: This paper addresses the important problem of avoiding input saturation when synthesizing control barrier functions (CBF) for safety specifications. It introduces a novel CBF formulation and a learning framework for learning to identify worst-case saturation states at the boundary of the safety set and improving the CBF. In addition, the paper includes many additional components to improve the efficiency and performance of the learning method, and demonstrates the efficacy of the proposed solution on a challenging pendulum-balancing domain with a simulated quadcopter. The paper does a sound job of mathematically motivating the design choices in this paper, and for these reasons I suggest this paper be accepted.

issues: * Additional experiments that include stochasticity to the dynamics of the simulated experiments would be more convincing for understanding whether the existing technique is potentially applicable to robot hardware. 
* A discussion on how challenging and important hyperparameter optimization is in this method for learning (e.g: including the regularization term, how many samples are warmstarted vs. taking from best of a batch, etc.)

Minor issues include:
* Replacing related works that are cited as being on arxiv with the peer reviewed versions (e.g: Clark, Andrew. "Verification and synthesis of control barrier functions." 2021 60th IEEE Conference on Decision and Control (CDC). IEEE, 2021.)
* The x and y axis titles on Figure 2 (b), © and (d) are quite small and enlarging them would improve readability
review_title: null

final_score: rating: Weak Accept: I recommend accepting the paper, but will not argue for my recommendation if the majority of other reviewers have a different opinion.
review_content: summary: This paper proposes a modification to the Control barrier function (CBF) formulation to allow for non-saturating controls, i.e. controls that do not exceed the actuation limits of the controller. The authors go about this by first characterizing a non-saturating invariant set, and then propose a learning method to learn the parameters of a control barrier function that will ensure a non-empty safe set and a resulting control law to keep the system within the safe set limits. Note that the authors assume that the dynamics of the system are known, and therefore this work is more in line with optimization based control rather than learning about the system dynamics. 
strengths_and_weaknesses: **Strengths:**
* Strong theoretical derivation and proofs ensure soundness of the work. While I am not aware of the full background of control barrier functions, I have checked the derivations and proofs within the paper. Prior knowledge was well referenced in the paper.
* Useful research direction: Controller saturation, and related simplistic modeling of actuation systems is a key limitation for a lot of simulation based robotics research and it is nice to see work addressing the issue.
* Challenging kinodynamic environment.

**Drawbacks:**
* Assumptions of knowledge of the system dynamics: As with any planning based approach, the requirement of knowing the system dynamics requires intensive efforts in system identification, and even then often the dynamics are only known with uncertainty. How senstive would the system be to misidentification of the dynamics? Perhaps the authors might be able to include an experiment on that.
* Affine control law: How frequently is the assumption of an affine control violated in practise? What are the typical ways in which this is addressed? 

Comments:
* The authors seem to have adopted an approach where the system starts with a saturating safe-set and works inwards from there. Would the authors comment on a reversed approach where they grow the safe set from a smaller starting safe set? I raise this question as even though the authors present results where the learned FI set was almost entirely safe, there were still saturated control inputs. 

robotics_focus: Highly relevant to robotics but no hardware experiments
quality_of_the_limitations_section: Limitations are addressed clearly
summary_of_recommendation: I found the presentation of the paper to be quite clear, the problem is well motivated and the research question is important. The prior works have been referenced, and the novelty of the paper is well established in the writing. I would recommend an accept, but as a non-expert in the field.
issues: Please refer to the main review.
review_title: null

final_score: rating: Weak Reject: I recommend rejecting the paper, but will not argue for my recommendation if the majority of other reviewers have a different opinion.
review_content: summary: this paper proposes a method to construct Control Barrier Functions using Neural Networks (NN) to avoid input saturations. The keypoint is to ensure the condition satisfaction on the boundary of the safe set, hence they formulate a min-max optimization problem with each step some counter-examples generated from the sampling process. As for the experiment, the authors design a 10d quadrotor-pendulum example and achieve better “satisfaction rate” and “forward invariant rollout rate” than non-NN CBF baseline approach.

strengths_and_weaknesses: Strengths: 
1. The paper is easy to read - it provides detailed thoughts on how to guide the training more efficiently and more stably.
2. Unlike papers working on only low-dimension systems, they work on a challenging 10d quadrotor-pendulum dynamic.

Weaknesses:
1. The approach is only testified on one experiment
2. The approach is only compared with one baseline
3. No hardware experiments or ablation studies (for different design choices and improvements)

robotics_focus: Sufficient demonstration on hardware
quality_of_the_limitations_section: Additional details required
summary_of_recommendation: I recommend rejecting this paper. Although the authors provide an interesting algorithm to construct CBF for input saturations, they lack thorough and solid experiments to persuade the readers about this approach’s benefit. The baseline is weak (only a CBF constructed in a non-NN fashion) and the experiment is only considering the case where an equilibrium point is available. The experiment cannot fully convince us why we need this new approach.

issues: 1. Weak baseline: I doubt the strength of this baseline. There are lots of related works [1-4] in using Neural Networks to construct CBF and have witnessed great success. Since NN is not a novelty of this paper, it is unfair to only compare a non-NN CBF. Besides, I would also suggest comparing methods such as reinforcement learning. 
2. Lack of experiments: The system used in the paper has an equilibrium point so the CBF is constructed following design-1. But the paper also introduces the design-2 which is not implemented in the experiment. More experiments are expected to also support that design choice.
3. Lack of ablation studies: We would like to see how each of the components / tricks in the paper leads to the performance gain.

[1] Wang, Chuanzheng, et al. "Learning control barrier functions with high relative degree for safety-critical control." 2021 European Control Conference (ECC). IEEE, 2021.
[2] Qin, Zengyi, et al. "Learning safe multi-agent control with decentralized neural barrier certificates." arXiv preprint arXiv:2101.05436 (2021).
[3] Meng, Yue, Zengyi Qin, and Chuchu Fan. "Reactive and safe road user simulations using neural barrier certificates." 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2021.
[4] Xiao, Wei, et al. "Barriernet: A safety-guaranteed layer for neural networks." arXiv preprint arXiv:2111.11277 (2021).
review_title: null

final_score: rating: Weak Accept: I recommend accepting the paper, but will not argue for my recommendation if the majority of other reviewers have a different opinion.
review_content: summary: This paper presents a framework for learning the Input Constrained Control Barrier Function (CBF) for high-dimensional nonlinear systems. The authors formulate a min-max problem to learn the non-saturating CBF. The maximization problem focuses on generating worst-case data points, while the minimization problem attempts to learn the CBF. The authors validate their approach in simulations and demonstrate that the proposed method achieves a higher success rate ensuring safety.
strengths_and_weaknesses: 1. Insufficient baselines. There are several works focusing on the input saturation problem using CBF, e.g. [1, 2]. The authors only compare with a baseline that does not consider input constraints. This would be difficult to evaluate the actual contribution. More comparisons with proper baselines would be desirable.
2. The interpretation of the CBF design is vague. There are several questions that are not well discussed. Why was the Design 2 introduced and how is this design used? How does it perform in terms of training and ensuring safety?
3. Some limitations are not discussed. 1. how to handle systems with higher relative order? 2. how to tackle problems with multiple safety specifications? Input constraints can also be considered as safety specifications.
4. In the proposed approach, the generation of counterexamples is crucial. However, it is difficult to assess whether the proposed generation technique can be generalized to different tasks, e.g., where the boundary is non-convex. Perhaps the authors could try another task with more complex safety boundaries, such as a robot manipulation task.
5. Several sections are not clearly explained. Some notations are used without explanation. 
6. Ablation studies are also beneficial to strengthen the paper.
robotics_focus: Highly relevant to robotics but no hardware experiments
quality_of_the_limitations_section: Limitations are not well addressed
summary_of_recommendation: Given the discussion in the strengths and weaknesses, I suggest that the authors should make a major revision. Improve the clarity of this paper, compare it with more appropriate baselines, and try different tasks to verify the generalizability of the proposed method.
issues: 1. What is $x_e$ in line 116? Why author introduces Design 2 which is not used in the experiment? 
2. In Figure 1, there is a noticeable difference between the learned set and the MPC set. 
3. The explanation of the analytic saturation risk is not very clear. What is the difference between $\mathcal{V}(\mathcal{U})$ and $\mathcal{U}$?
4. “Another challenge to speed is systems with large state spaces…” 
5. “We project the saturation risk’s gradient along the boundary, use this to update x, and finally project x back to the boundary.” How is the projection done? It would be beneficial to visualize the procedure to improve the understanding.
6. What is $s_{\theta}^*$, and how is it computed? What are $\mathbb{X}_{reg}$ and $\mathcal{D}$?

[1] Agrawal, Devansh R., and Dimitra Panagou. "Safe control synthesis via input constrained control barrier functions." *2021 60th IEEE Conference on Decision and Control (CDC)*. IEEE, 2021.

[2] Garg, Kunal, and Dimitra Panagou. "Control-lyapunov and control-barrier functions based quadratic program for spatio-temporal specifications." *2019 IEEE 58th Conference on Decision and Control (CDC)*. IEEE, 2019.

[3] Dalal, G., Dvijotham, K., Vecerik, M., Hester, T., Paduraru, C., & Tassa, Y. (2018). Safe exploration in continuous action spaces. *arXiv preprint arXiv:1801.08757*.

[4] Liu, P., Tateo, D., Ammar, H. B., & Peters, J. (2021). Robot reinforcement learning on the constraint manifold. In *Conference on Robot Learning* (pp. 1357-1366). PMLR.

[5] Akametalu, Anayo K., et al. "Reachability-based safe learning with Gaussian processes." *53rd IEEE Conference on Decision and Control*. IEEE, 2014.

[6] Bansal, Somil, et al. "Hamilton-jacobi reachability: A brief overview and recent advances." *2017 IEEE 56th Annual Conference on Decision and Control (CDC)*. IEEE, 2017.
review_title: null

Meta-review: metareview: The authors propose an interesting work building on the control barrier functions.
The methods seem to be sound and the experiment in a high-dimensional setting, the quadrotor one, seems to be very convincing.
The work requires the knowledge of system dynamics: while this is a limitation, it is a common assumption of a robotics system and doesn't invalidate the importance of the proposed work. The reviewers ask for a bit of discussion on the topic, in particular focusing on the possibility of an inaccurate model and how this will impact the learning system.

The main issues of this paper are the lack of citations of recent methods in safe reinforcement learning and the lack of baselines in the experimental evaluation.

For the acceptance of this paper, the authors must update their paper with all the suggested citations from the reviewers and add additional baselines to the experimental evaluation.

Reviewers suggest also adding some more ablation studies to show the robustness of the approach. This would strengthen the quality of the work.

A minor issue of this work is the lack of clarity in some notation. Authors should make sure that the notation is introduced properly.

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The reviewers agree on the fact that the paper is interesting and positively value the novelty of the approach. Most reviewers lean towards acceptance of the paper. Unfortunately, one reviewer is not convinced about the strength of the current experimental evaluation, in particular the lack of appropriate baselines.

While I think that the reviewer is right in highlighting this drawback, I think the authors should be able to implement the baseline for the final version of the paper. Indeed, the proposed task is quite complex, thus making it difficult to implement such a baseline in the restricted timeframe of the rebuttal.
Furthermore, the authors put reasonable effort into the rebuttal phase.

This paper is clearly borderline, but it may be worth considering accepting it, at least for a poster presentation, if the authors are able to complete the required baselines for the camera-ready version. 

A minor, but important, note: in case this paper is accepted, the authors must fix the formatting. In the latest version, the appendix appears before the bibliography, but this should not happen in the camera-ready.
confidence: 2: The area chair is not sure
Decision: Accept (Poster)