__ Summary and Contributions__: This paper essentially claims two contributions: (1) it proposes to do randomized smoothing on soft confidence scores rather than on hard classifications, and (2) it presents a CDF-based robustness certificate (applicable to randomized smoothing on soft scores but not hard labels) that is potentially tighter than the standard expectation-based robustness certificate. I like contribution (2), but I am less excited about contribution (1), because soft smoothing was already discussed in [https://arxiv.org/abs/1906.04584].

__ Strengths__: The paper proposes an interesting randomized smoothing certificate that is based on the distribution of confidence scores rather than on just their expectation. Typically, in randomized smoothing, one is told the expectation, over random Gaussian noise, of f(x+noise), and the goal is to lower-bound the expectation of f(x'+noise), where x' is a nearby point and the output of the function f is assumed to be bounded -- say between 0 and 1. The "worst case" f, which attains the lower bound, is a piecewise constant function with steps orthogonal to the perturbation (x' - x) which puts 1's on the side closer to x and 0's on the side closer to x'. The technical observation in this paper is that we can do better if we are told not just the expectation of f(x+noise) but its CDF. (Like the mean, the CDF can be estimated from samples.). Intuitively, the "worst case" f from the original bound has the distribution of f(x+noise) being very anti-concentrated, so if we are told that, to the contrary, this distribution must be concentrated, then that rules out the original bound's worst case. The new bound proved in this paper is similar to the old bond, in that the "worst case" base classifier is still a piecewise constant function with steps orthogonal to the perturbation (x' - x). For example, we are told that P[ f(x+noise) < 0.4 ] = 0.8 and P[ f(x+noise) < 0.6] = 1.0, then the new "worst case" f is a piecewise constant function which puts 0.6's on the side closer to x and 0.4's on the side closer to x'.

__ Weaknesses__: The weaknesses of this paper mostly have to do with the motivation of confidence smoothing:
-- Confidence smoothing was already discussed in Salman et al (https://arxiv.org/abs/1906.04584) under the name "soft smoothing"; your Theorem 1 is an immediate application of their Lemma 2.
-- As the authors write, deep network classifiers are not well-calibrated, so it's unclear how valuable it is to have the confidence information.
-- The plot in Figure 4 looks to me like it does show a correlation between prediction score and certified radius.
The paper would be stronger if you could demonstrate that confidence smoothing with the CDF-based certificate outperforms hard label smoothing with the standard expectation-based certificate. As is, you do show that confidence smoothing with the CDF-based certificate outperforms confidence smoothing with the expectation-based certificate, but that isn't a reason to use confidence smoothing in the first place.

__ Correctness__: I think so.

__ Clarity__: Yes.

__ Relation to Prior Work__: Yes, except the authors do not discuss "soft smoothing" from Salman et al.

__ Reproducibility__: Yes

__ Additional Feedback__: Update: I have read the rebuttal, and I'm keeping my score. I think that the motivation is a little bit weak, given the prior Salman et al. paper, but this submission is possibly over the bar.
I just realized that Lemma 2 from Salman et al. only covers the case where \sigma = 1. That said, here is how to use Lemma 2 from Salman et al. to prove your Theorem 1, in the special case where \sigma = 1.
Lemma 2 from Salman et al states that if g is any function whose output is bounded in [0, 1] then the function \Phi( \bar{g} ) is Lipschitz with constant 1, which implies:
\Phi^{-1}( \bar{g} (x') <= \Phi^{-1}( \bar{g} (x')) + |||x' - x||
Applying the monotone function \Phi to both sides of this inequality yields:
\bar{g} (x') <= Phi(\Phi^{-1}( \bar{g} (x')) + |||x' - x||)
Now, to prove your Theorem 1 using this result, we define the function g(x) = (f(x)-a)/(b-a), which is bounded in [0, 1] as required. We then apply that lemma, then multiply both sides by (b-a), then add a, and then rearrange, to obtain the first statement of Theorem 1. (The second statement can be proved in the same way.)
To generalize to the case where \sigma is not 1, you could invoke Lemma 1 from Levine et al (https://arxiv.org/abs/1905.12105), which is the exact same thing as the Salman et al lemma, but for general \sigma.

__ Summary and Contributions__: This paper investigates certified defense regarding confidence scores (rather than mere classification) via randomized smoothing. By considering *confidence distribution*, the resulting method (theory) beats the baseline method only using the confidence average.

__ Strengths__: In my opinion, confidence certification is very important when deploying machine learning models in real-world applications. The theoretical result is non-trivial, behind which the intuition is well explained.

__ Weaknesses__: See additional feedback for several flaws (or suggestions).

__ Correctness__: The claims and method make sense to me.

__ Clarity__: Yes, the paper is well written and easy to read. The background and the proposed method are clearly presented.

__ Relation to Prior Work__: Related work is introduced from the randomized smoothing view. It would be better to also talk about whether there are privious work focusing on confidence score certification.

__ Reproducibility__: Yes

__ Additional Feedback__: ## After reading the feedback
The feedback about parameters s1,..., sn is satisfactory. I expect the discussion is added to text with more details, maybe experiments investigating the parameters.
***
* The parameters s_1… s_n are not discussed in text. It would be better to talk about how to set these parameters, and how these parameters affect the performance.
* 240: “n = 100,000” should be “m = 100,000”.
* I would like to suggest to add experiments on ImageNet — only one dataset (CIFAR-10) is not sufficiently convincing.
* It would be better to add discussion about related work on certifying confidence.

__ Summary and Contributions__: Randomized smoothing generates a certified radius for a classifier’s prediction without conveying any information about how confident an input point should be in the assigned label. This work proposes to address this problem by restoring the confidence information in certified classifiers and further averaging the confidence scores (measured by average prediction score and the margin) from the underlying base classifier for prediction.

__ Strengths__: - The motivation of measuring confidence is clear.
- Theorems are carefully proved.
- The confidence measure helps the performance.

__ Weaknesses__: - Some model design rationale is unclear.
- Experimental studies relying on single dataset may not be convinced.
- It lacks detailed experimental analysis.

__ Correctness__: The authors provide a working and correct solution for the problem, which seems technical sound.

__ Clarity__: The presentation is good and clear.

__ Relation to Prior Work__: The paper is missing a section discussing related works regarding randomized smoothing. It’s strongly suggested to include a detailed discussion how the contribution made by this paper differs from prior works.

__ Reproducibility__: Yes

__ Additional Feedback__: - In page 3, how to determine the parameter a, b and c?
- The paper uses two notions to measure confidence, i.e., the average prediction score, and the margin. The definition of these two notions are straightforward; the rationale of such design rational is not explicit and sound – i.e., why these two notions can be used for confidence measure.
- The current experiments are conducted using ResNet-110 on CIFAR-10. It may not be sufficient. The results on ImageNet are also recommended as Cohen et al. [6] did.
- What is the relationship between radii \sigma and confidence threshold? Why the authors select different thresholds for \sigma=0.25 and 0.5 in Figure 2?

__ Summary and Contributions__: The authors extend the idea of providing certified radii around input points for class assignment to the provision of confidence.
This deals with the issue that, just because it is confidently not an adversarial example doesn't mean it's not very uncertainty about the actual class.
They use the underlying base classifier's confidence scores at the samples and use this to determine a bound on the capacity for an adversary to manipulate it.

__ Strengths__: Uncertainty quantification is often forgotten in ML work. This seems a useful paper which allows users of such tools to also have access to the confidence of the classifier - this is something a user might use to make decisions with and so it is valid that it also should be protected from adversarial attack. I am not aware of work quite on this topic & so it has novelty.
The claims (and theorems) appear to be sound.
I like the empirical experiments - in particular the check to see what relationship exists between the softmax confidence score and the radius.

__ Weaknesses__: I can't see serious weaknesses.
Maybe application to another dataset?
Also could this be applied to other classifiers (not DNNs?)

__ Correctness__: I'm not very confident - but I think they are.

__ Clarity__: I also like the number of examples used throughout.
It is clear prose. The mathematically lemmas are accompanied by coherent proofs.

__ Relation to Prior Work__: It seems to discuss the field sufficiently to place itself as a novel contribution.
There is some work by Kathrin Grosse (e.g. "The Limitations of Model Uncertainty in Adversarial Settings") that concerns itself with uncertainty I think?

__ Reproducibility__: Yes

__ Additional Feedback__: line 80: typo "withing" -> with in
it might be nice for the figure numbers to be in the order of the text?
line 193: The equations surely should be mu_X and mu_Y in the ratios.
===Having read the other reviews and feedback===
I have low confidence in this area. However, for me the two main issues are now:
1) The need for an additional experiment (as suggested by the other reviewers).
2) There seems to be literature that has been missed by the authors (e.g. that mentioned by reviewers 1 & 3. including Salman et al...)
However:
I feel that these can be added as requirements for the changes after submission, for the camera ready version however, so I'm happy to leave my score the same.