From: ICLR.cc/2023/Conference/Paper250/Authors
title: Submission Withdrawn by the Authors

From: ICLR.cc/2023/Conference/Paper250/Authors
title: Response to Reviewer jrgh
comment: We thank you for your helpful comments based on your in-depth understanding of our paper.

>Regarding the comparison with the SOTAs DDRS and DSRS

Thank you for your suggestion. DDRS and DSRS had not provided the code when we submitted this paper. Thus, we cannot compare the time cost directly. However, we can briefly analyze their computational complexity. For the proposed method and DDRS, the former uses an algorithm of $\mathcal{O}((\frac{1}{2})^t)$ convergence rate, and the latter uses an algorithm of $\mathcal{O}(\frac{1}{t})$ convergence rate (assume gradient descent with $L$-smoothness). In addition, DDRS maintains a memory bank and uses back-propagation several times, which is computationally expensive.
Thus, the proposed method cost less to optimize sigma compared to DDRS.
On the other hand, the running time of DSRS is roughly the same as Cohen’s method according their paper. In this paper, we show that the proposed method takes about 7% additional inference time. Thus, it is also roughly the same as Cohen’s method. 
We add the time cost analysis in Appendix A.2.


> The objective being strict quasiconcavity indicates that it is unimodal...

Gradient-based methods may perform well; however, if we consider the convergence guarantee and the convergence rate, QCRS will be much better. 
Gradient-based methods will fail in the quasiconcave curves since there are many small-gradient or zero-gradient points. As shown in Figure 2, there are many zero-certified-radius points, and those points also result in zero gradient, making GD fail.

The proposed QCRS uses a looser assumption than the ones used gradient-based methods.
In addition, it provides a much faster convergence rate ($\mathcal{O}((\frac{1}{2})^t)$) to achieve $\delta$-optimal, while gradient-based methods do not provide any guarantee.
Therefore, the proposed method is superior to GD-based methods such as DDRS (Alfarra et al. 2022). 
Although the ACR improvement of QCRS w.r.t. DDRS in Table 2 is not significant, QCRS is much faster and provides a convergence guarantee.



>Convergence analysis of Algorithm 1

Thank you for your constructive suggestion. We add a new section (Appendix A.1) to analyze the convergence of QCRS. Algorithm 1 converges if sigma-SQC-condition holds.

From: ICLR.cc/2023/Conference/Paper250/Authors
title: Response to Reviewer 96UE
comment: We thank you for your valuable comments based on your expertise.

>Regarding the gradient descent (Weakness 1): 

GD will fail in the quasiconcave curves since there are many small-gradient points or stationary points. As shown in Figure 2, there are many zero-certified-radius points, and those points also result in zero gradient, making GD fail.
In addition, as mentioned in the general response, the gradient descent method needs $L$-smoothness and $\mu$-strong concavity to guarantee a convergence rate.
The proposed method only needs a looser condition (quasiconcavity) to converge, and the convergence rate is $\mathcal{O}((\frac{1}{2})^t)$.
Although gradient descent may not fail, the convergence is not guaranteed, not to mention the convergence rate.
In addition, back-propagation in gradient descent methods is unstable and time-consuming due to MC sampling.
Thus, considering speed and convergence, we believe QCRS is  better than the gradient-descent-based methods used in prior work.

 
>Regarding the curves in Figure 2 (Weakness 1). 

Figure 2 is based on standard RS (Cohen). We only consider standard RS in this paper.
We sample 20 sigma values to plot Figure 2, listed below: 
0.15, 0.18, 0.2 , 0.21, 0.22, 0.23, 0.24, 0.25, 0.26, 0.27, 0.28, 0.29, 0.3, 0.31, 0.32, 0.33, 0.35, 0.4, 0.45, 0.5.
Because the model in Figure 2 is trained using sigma=.25, the valid sigma values (those can produce a positive certified radius) should be around 0.25. Thus, we increase the sampling density around sigma=.25 to check the quasiconcavity.
Regarding Figure 2, we use numerical measurement to verify the quasiconcave condition (according to Lemma 2).
Since we want to achieve the $0.01$-optimal sigma, we check the quasiconcavity based on the points on the $0.01$-grid (gird with $\delta=0.01$ line-to-line spacing).
Therefore, we sample sigma in the step size of 0.01.
If we decrease $\delta$ to check quasiconcavity, the $\delta$-optimal optimization becomes more accurate but the quasiconcave condition is stricter. There is a trade-off to choose $\delta$.
We have added explanation regarding Figure 2 in Apppendix. 
 

>Regarding the trade-off issue 

We can observe two kinds of trade-offs in standard RS. One is in the training phase (different sigma values are used to train the base classifier), and the other is in the inference phase (different sigma is used to build the smoothed classifier). The trade-off that you mentioned in Figure 4 results from training since those three figures indicate three different models trained by different sigma values. In this paper, we improve RS in the inference phase. Thus, the trade-off issue we overcome is in the inference phase. When we use different sigma to inference, we can observe a trade-off in the radius-accuracy curve. The proposed method can increase the ACR, overcoming the trade-off issue in the inference phase. 
We have rephrased the sentence in the revised version to avoid any misunderstanding.

From: ICLR.cc/2023/Conference/Paper250/Authors
title: Response to Reviewer n5tN
comment: We appreciate your careful reanding and constructive comments.

>Regarding theoretical justification 

The sigma-SQC condition (Definition 2) is our data assumption.
We propose this condition based on our observation of the sigma-radius curve.
We evaluate the sigma-SQC assumption on CIFAR-10 and ImageNet, observing that 99% of datapoints satisfy this assumption. 
Regarding theoretical justification, we can prove the convergence of Algorithm 1 using this sigma-SQC assumption.

>Regarding regression/forecasting setting

This paper aims to improve the RS certified region on classification tasks. The regression task is beyond the scope of this study but is an interesting future work.

>Regarding unclear writing. 

We apologize for the unclear writing. We have rephrased some sentences. We will add the strict definitions of radius (certified region), sigma-radius, sigma-radius curve, and truncation bound in the next version. 

>Regarding the fairness and dimension issues

The fairness and dimension issues are the well-known difficult challenge for RS. They are interesting but not the issue that this paper wants to discuss. This paper mainly focuses on building the better smoothed classifiers to improve the certified region. 

>Regarding the novelty of the optimization method

As you say, the optimization method in this paper may not be novel. However, we are the first to combine quasiconcave optimization and the bisection algorithm to optimize the sigma values for RS, and the experiments show that the proposed method is efficient and effective. We analyze the convergence and conduct many experiments to show that we can improve certified radii using this simple and efficient optimization algorithm. 


>Regarding Definition 2 (sigma-SQC condition)

In this paper, we discuss the quasiconcavity of certified radii w.r.t. sigma.
We have rephrased some sentences to emphasize this.
In addition, in Definition 2, 2 on RHS illustrates those two probability terms ( $\mathop{Pr}\limits_{\sigma < \sigma^* }[\nabla R (\sigma)>0]$ and $\mathop{Pr}\limits_{\sigma > \sigma^* }[\nabla R (\sigma)<0]$ ) in the LHS are close to one, so the summation is 2 on the RHS.

 

>Regarding different class of models 

Thank you for your suggestion. It is unclear what will happen if considering transformer or RNN in RS. We will delve into this in future work. In this paper, we focus on Cohen’s RS. The proposed method is based on their model and MACER models.

From: ICLR.cc/2023/Conference/Paper250/Authors
title: Response to Reviewer pGBY
comment: We would like to thank you for your helpful comments.

>Regarding the concern about contribution (Weakness 1). 

The concave assumption is very strong, and almost no datapoint satisfies this assumption. 
In contrast, the proposed method uses a much looser assumption – quasiconcavity. 
On the other hand, the previous gradient-based works (DDRS) cannot guarantee convergence when optimizing sigma, but the proposed method can. This transformation is necessary. The assumption is looser, and the convergence rate is faster (from no convergence guarantee to O((1/2)^t)). 

 

>Regarding the missing baselines (Weakness 2). 

Thank you for your suggestion. To the best of our knowledge, we can improve RS in the training or inference phase. The two works that you mention are training-based methods, and the proposed QCRS is an inference-based method. That is, the proposed method does not utilize any training, so it is unfair to compare QCRS with those training-based methods directly. For fair comparisons, we mainly compare the proposed method with inference-based RS methods.


>Questions about the time-consuming and three specified constant sigmas for ACR. 

We retest the time cost experiments. The overhead cost is roughly 7% (6.5 sec. to 6.96 sec.). MC sampling (100k points) takes over 90% of running time when inferencing. Calculating specific sigma in Algorithm 1 only takes a small portion of running time.
Regarding three specified constant sigmas, since the proposed method is inference-based and we do not train any new model, we use the weight provided by Cohen and MACER. Those three specified constant sigmas are from their settings.

From: ICLR.cc/2023/Conference/Paper250/Authors
title: Response to Reviewer ousJ
comment: Thank you for your valuable comments. Following your comments, we revised our paper and answer the questions as follows:

>How general this quasiconcave condition is? 

Quasiconcavity is the data assumption in this paper. We show that if the quasiconcave condition holds, the proposed method guarantees to find the optimal sigma value (to build the best smoothed classifier). In addition, to evaluate how general this condition is, we randomly select several samples from the real-world datasets (CIFAR-10 and ImageNet) and check their quasiconcavity. The results show that roughly 99% of data satisfy this condition. 

>Regarding novelty 

Prior work did not discover any property to optimize the smoothed classifiers and thus did not provide any guarantee to find the optimal sigma. In contrast, we delve into the sigma-radius relationship and propose the sigma-SQC condition (much looser than what prior work needs). This is the assumption we use to design our QCRS method. We find that roughly 99% of datapoints in CIFAR-10 and ImageNet satisfy this assumption. This is why QCRS can  significantly improve the certified region with little computational overhead. 

>Regarding reproducibility

Code will be made available after acceptance. 

>There are a few places that require some clarification... (Lemma2) 

We apologize for this misleading notation. “f” indicates the classifier in this paper, and in lemma 2, “f” is an arbitrary function that is used to illustrate quasiconcavity. We replaced "f" with "h" in Lemma 2. 

>Regarding Definition 2 

Definition 2 is based on Lemma 2, which shows the necessary condition for the quasiconcave condition in RS (i.e., sigma-SQC). That is, positive gradient on the left hand side of the optimal solution and negative on the right hand side (probability = 1 for both terms, so the summation is 2).
Definition 2 (sigma-SOC condition) is our data assumption, which guarantees that Algorithm 1 finds the optimal radius.
It is unnecessary to verify this condition before RS. If the condition does not hold, the corresponding RS still works but does not guarantee the optimal radius. In contrast, if it holds, the optimal radius is guaranteed.

From: ICLR.cc/2023/Conference/Paper250/Authors
title: General response
comment: We would like to thank all the reviewers for their valuable and constructive comments. Based on these comments, we revised our paper and submitted a revision. In the main paper, following the policy of ICLR, we slightly rephrase some sentences and context according to reviewers' comments. In addition, we add some new sections in the Appendix, such as convergence analysis and time cost analysis. 

Following reviewers' comments, we first reply to common questions as follows:
### The simga-SQC condition
In this paper, we consider certified radii with respect to sigma, observing the quasiconcavity, and propose the simga-SQC condition. Note that the simga-SQC condition is our data assumption. That is, if a datapoint satisfies this condition, Algorithm 1 guarantees that the optimal sigma would be found. However, if this condition does not hold for a datapoint, the RS of this datapoint still works, but the smoothed classifier may not be optimal. We use statistical analysis on the real-world dataset to verify the generality of the sigma-SOC condition, and roughly 99% of the images from CIFAR-10 and ImageNet satisfy this condition. 

### Time cost
Regarding the experimental results of time cost, we rerun the experiments and found that the previous numbers in Table 1 were not very accurate due to the interference from other running jobs on our device. Thus, we prepare a single independent dedicated PC (RTX3090+AMD5600X+32GB-DRAM) to run the time cost experiments again.
The results are as follows:
-|Time cost (sec.)
---|:---:
Cohen|6.5
Grid search|155.8
QCRS|6.96

The computational overhead is 7%, which is higher than the original results but is still relatively small compared to other methods. We illustrate the detailed setting in Appendix A.2. 

In addition, we briefly analyze the computational cost compared with standard RS.
The sigma searching region of Algorithm 1 is $0.5-0.12=0.38$.
Because the convergence rate of Algorithm 1 is $\frac{ \sigma_{max} - \sigma_{min}}{2^t} \geq | \sigma_t - \sigma^*|$, if $t \geq 6$, we can achieve $0.006$-optimal (i.e., $|\sigma - \sigma^*| < 0.006$, because $0.38/2^6=0.0059$).
For each iteration in Algorithm 1, we need to compute $1,000$ forwards.
Thus, for each datapoint, we roughly need additional $6,000$ forwards.
The standard RS needs $100,000$ forwards, so the overhead of the proposed QCRS is 6%.
We also analyze the computational cost of DSRS and DDRS in Appendix A.2.

### Novelty
Our proposed QCRS method is novel in that it requires a much looser assumption (sigma-SQC) and obtains the optimal sigma with a much faster convergence rate than prior work.
The below table summarizes the comparison to DDRS (Alfarra et al.), a gradient-based SOTA method for finding the optimal sigma.

-- |QCRS|DDRS
---|:---:|---: 
assumption to achieve the optimal|$\sigma$-SQC|concavity
generality (% of data that satisfy the assumption)|high (99%)|low (7%)
convergence|$\mathcal{O}((\frac{1}{2})^t)$|no
back-propagation|no|yes
Time cost| low | high

Compared to DDRS, the proposed method uses a much looser assumption, and the convergence rate is $\mathcal{O}((\frac{1}{2})^t)$. DDRS (gradient-descent-based) is based on the concave assumption (stricter than quasiconcavity), and this assumption alone cannot  guarantee any convergence for $\delta$-optimality. Only if both $L$-smoothness and $\mu$-strong concavity hold, the gradient-descent-based methods can provide $\mathcal{O}((\frac{L-\mu}{L+\mu})^t)$ convergence.

From: ICLR.cc/2023/Conference/Paper250/Reviewer_ousJ
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: This paper proposes a quasiconcave optimization approach to find the optimal certified radii for RS. The authors claim and demonstrate most date points satisfy this extra quasiconcave condition. Experiments are provided to demonstrate the competitive performance of the proposed approach on Cifar10 and ImageNet. 
strength_and_weaknesses: Strength:
  1. The proposed algorithm seems to be easy to scale.
  2. The results on CIFAR10 and ImageNet seem to be promising.

Weaknesses:
  1. It is still unclear how general this quasiconcave condition is. The authors show that such a condition seems to be useful for existing benchmark problems. I am not sure how general such a condition is. It will be much more useful if the authors can provide some rigorous proofs for such a condition to hold under certain data assumptions.
  2. It is quite difficult to reproduce any of the results in the paper.
clarity,_quality,_novelty_and_reproducibility: Most parts are quite clear. There are a few places that require some clarification. For example, is the output of f a scalar or a vector? I mean, e_i seems to be a vector. But Lemma 2 assumes the scalar case?

It is also very difficult to judge the true novelty of the work. Definition 2 is hard to understand. This sigma-SOC condition seems to jump out of nowhere. How is such a condition "exactly" verified? Does one have to first verify this condition and then do RS? 

Regarding reproducibility, it seems that no code has been provided. It is very difficult to reproduce any of the results. 

summary_of_the_review: I cannot recommend acceptance due to two main issues: 1. The true novelty is not clearly justified in a mathematically rigorous way. 2. It is very difficult to reproduce the results. 
correctness: 3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
flag_for_ethics_review: NO.
recommendation: 5: marginally below the acceptance threshold

From: ICLR.cc/2023/Conference/Paper250/Reviewer_pGBY
confidence: 5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.
summary_of_the_paper: In order to provide a sizeable certified radius for real-world datasets, one way to solve this is to use an input-specific \sigma to calculate the radius for each sample. The paper discovers that the radius curves are quasi-concave for most data points. The authors propose an input-specific algorithm QCRS utilizing dichotomy to find the optimal \sigma for each input. 
strength_and_weaknesses: Strength:

1. QCRS solves the problem of the truncation effect, which is a severe problem for original randomized smoothing proposed by Cohen et al. 

2. The experimental results are substantial compared with Cohen et al. 

Weakness:
1. The main concern is that the technique seems trivial. Alfarra et al. already assume that sigma-radius curves are concave and use gradient-based optimization to find the suitable \sigma. Even though the authors claim that most curves are not concave but quasi-concave, I thought this transformation did not contribute much. QCRS can also work with concave assumptions. 

2. Experiments: Several important baselines are missing. The paper cites Jeong et al., 2021 but does not compare with it, which is an essential randomized smoothing method based on adversarial training. Furthermore, the paper also misses recent baselines like [1][2].

3. Questions: 
(a) Calculating specific \sigma for each data point is time-consuming; why does QCRS only raise 0.05 seconds to Cohen et al.? 
(b) If QCRS aims to find \sigma to get the largest robust radius, why \sigma calculate for ACR started by three specified constants? 

[1] Salman et al., Provably Robust Deep Learning via Adversarially Trained Smoothed Classifiers. in NeurIPS, 2019.

[2] Jeong et al., Consistency regularization for certified robustness of smoothed classifiers. in NeurIPS, 2020.
clarity,_quality,_novelty_and_reproducibility: The paper is easy to follow. However, the novelty is limited, as I mentioned above. 
summary_of_the_review: The paper is based on previous work, and the contribution is minor. The experiment misses some important baselines. Thus I vote for rejection. 
correctness: 3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 5: marginally below the acceptance threshold

From: ICLR.cc/2023/Conference/Paper250/Reviewer_n5tN
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: This work propose a way to find the parameter of RS efficiently, based on the assumption and idea of quansi-concavity.
strength_and_weaknesses: ## Strength
- draw good observation on quasi-concavity of the parameter and derive proper bisection algorithm for searching
- extensive experimetns on classification setting


## Weakness
- no theoretical justification on the assumption of quasi-concavity or any strong intuition, similarly in the summary of contribution, the author did not prove anything, rather say 'empirical confirm'
- there are other class of regression problems recently being adopted beyond classification, see https://arxiv.org/abs/2202.11910 and https://arxiv.org/abs/2207.09572. How this method improve regression/forecasting settings empirically and theoretically? 

clarity,_quality,_novelty_and_reproducibility: - several unclear writing in the draft
 - in the introduction, some terms like radius, sigma-radius, sigma-radius curve and truncation bound are not well-defined. If all of terms or jargons are going to be describe, please indicate proper section or paragraph. 
 - weak connection between the challenge mentioned in the intro and the propose method. For example, the paper did not cover fairness or dimension issues
 - the algorithm is efficient based on quasi-concavity, but cannot be claimed to be novel and that is standard in optimization literature and other methods also exist
 - what does it mean by 'the curse of dimensionality for input-dependent smoothing and provided a practical input-specific method to deal with that issue.'?
 - clearly descirbe (quasi)concave w.r.t. what over what in the draft? I believe the answer is sigma.
 - need more detailed discussion on the SQC definition. why author consider this definition and connection with quasi-concave? why 2 on RHS?  

summary_of_the_review: Marginal contribution on the literature but not novel. To claim more novelty, requires theoretical justification or more. To claim empirical contribution, recommend to consider regression settings and others, along with more diverse setups like different class of models, e.g., RNN, CNN, Transformer, and even comparing with other defense mechanism for the references, like minimax defense. The paper writing can be more enhanced with more clear descriptions and linking motivation and actual things covered in the papers.
correctness: 3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
details_of_ethics_concerns: N/A
recommendation: 3: reject, not good enough

From: ICLR.cc/2023/Conference/Paper250/Reviewer_96UE
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: This paper focuses on Randomized Smoothing used in Robustness Certification. Classical works obtain optimal radius to each input with a fixed $\sigma$, and recently, some works propose a method to maximize the certification radius to each input with different $\sigma$. They use a gradient-based optimization to optimize it, which assumes that the curve is concave. The authors show the certified radius curves empirically evaluated over 200 samples in CIFAR-10, and find most of the curves are not concave but quasi-concave. Based on this,  they propose a method to optimize the problem without a gradient-based optimization. They show that their method is both effective and efficient and verify the superiority of the proposed method in CIFAR-10 and ImageNet.
strength_and_weaknesses: Strength
1. Randomized smoothing is a popular way to obtain the robust radius of a sample with a theoretical guarantee recently, and analyzing this problem from a Quasi-concave optimization perspective is new for me. 
2. The method doesn't use backpropagation to calculate the gradient and is more efficient than previous work. 
3. The authors also evaluate the proposed method with sufficient experiments.

Weakness
1. Although the authors show the sigma-radius curve is not concave, Gradient descent may rarely fail based on Figure 2. The experiment setting in Figure 2 is not clear, the authors should explain it in detail.
2. It is unclear why QCRS has a similar time cost with COHEN, because QCRS needs more steps to find the optimal $\sigma$ for every sample.
clarity,_quality,_novelty_and_reproducibility: 1. In Figure 2, every curve seems to be piecewise linear, is it because the authors just approximate it using a few sigmas which is not enough? I suggest the authors explain it. Because this is important to distinguish whether the curve is quasi-concave. Besides, which algorithm to evaluate the certified radius in Figure 2? Whether a different algorithm makes an impact on the shape of the curve?
2. Gradient descent(GD) is guaranteed to converge to a stationary point of the function($\nabla f(x) = 0$). Because a stationary point doesn't have to be a local minimum, GD would fail. Thus, the key point may be not whether the curve is concave or quasi-concave, but whether they're stationary points that are not local minima. In Figure 2, it seems to be not clear, and I suggest the authors show why GD could fail in this problem. 
3. In Table 1, is the "Time cost" means the time cost of evaluating one sample on average? It is amazing that QCRS is so close to COHEN, because QCRS needs more steps to find the optimal $\sigma$ for every sample, why it could be so close to COHEN? 
4. The authors say "Our method overcomes the trade-off between clean and robust accuracies on the radius-accuracy curve". But in Figure 4, it is not clear why it overcomes the trade-off. Because the trade-off still exists in the proposed method.

summary_of_the_review: The authors proposed a novel algorithm named QCRS to calculate the robust radius. They empirically show the sigma-radius curve is usually not concave but quasi-concave. They combine quasi-concave optimization and the bisection algorithm to obtain an effective and efficient method. Various experiments show the superiority of the proposed method.
correctness: 3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
flag_for_ethics_review: NO.
recommendation: 6: marginally above the acceptance threshold

From: ICLR.cc/2023/Conference/Paper250/Reviewer_jrgh
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: This work exploits the quasiconcavity of the sigma-radius curve in randomized smoothing. The authors define a $\sigma$-SQC condition for the sigma-radius curve, which indicates strict quasiconcavity, and numerically show that the quasiconcavity commonly exists in the sigma-radius curve of real world data points. Based on this observation, the authors propose QCRS, a bisection-type randomized smoothing method which assumes and utilizes the quasiconcavity of the sigma-radius curve. Extensive experiments demonstrate the improved certified region of QCRS compared with existing methods.
strength_and_weaknesses: Strength:
- Simple and intuitive strategy which shows improvement with little running time overhead.
- The presentation is pretty good. The authors did a good job explaining the mechanism and comparison with the SOTAs. 

Weaknesses:
- Most of the empirical comparison is devoted to the baseline COHEN, while the comparison with the SOTAs DDRS and DSRS seems insufficient. The authors claim that QCRS is more time-efficient than them, and that the way QCRS approximates the gradient sign is better than directly using the unstable gradient in the previous methods. Although these points sound reasonable, some running time comparison with DDRS and DSRS is necessary. The authors only compared the running time with Grid search and COHEN. 
- The objective being strict quasiconcavity indicates that it is unimodal. In this case, gradient-based method should perform reasonably well. I would like see stronger empirical evidences that Algorithm 1 is better than the gradient-based strategy in Alfarra et al. (2022).  In fact, the improvement of QCRS w.r.t. DDRS in Table 2 is not significant if the randomness is considered.
- It would be great if the authors can analyze the convergence of Algorithm 1, since strict quasiconcavity is already a very nice property for convergence analysis.
clarity,_quality,_novelty_and_reproducibility: This paper is well-written and has a nice flow of ideas.
summary_of_the_review: This paper propose a simple and effective strategy (QCRS) to improve the certified robustness region of randomized smoothing. However, more empirical comparison with the existing methods, especially DDRS (Alfarra et al., 2022), is necessary to further demonstrate the benefits of QCRS.
correctness: 3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 5: marginally below the acceptance threshold