Certified Defense Against Complex Adversarial Attacks with Dynamic Smoothing
AI safety;adversarial robustness;randomized smoothing
Yes
No ethics review needed.
1. Could the authors provide the evaluation results of their method on the entire CIFAR-10 test set? Using the full test set is standard in the field and would provide stronger evidence for the method's generalizability.


2. The authors exclude baselines based on $l_{p}$-norm defenses with $p > 2$. Could the authors provide a more detailed justification for this choice? Although this is briefly mentioned in Section 5.1 under **Baselines**, it is not very clear to the reader why this choice was made. Specifically, could the authors expand on the line: *“We do not consider randomized smoothing techniques with certification guarantees in terms of $l_{p}$ norms with $p > 2$, since impossibility results are known for increasing $p$.”*?


3. Under the multi-attack setting, only one combination of attacks (Square Attack + FGSM) was evaluated. Testing multiple attack combinations could better demonstrate DSMOOTH's robustness. Are there plans to expand the experimental setup to include additional attack combinations?


4. Other works have discussed anisotropic approaches in the context of randomized smoothing, such as Hong and Hong (2022) and Eiras et al. (2021). Could the authors elaborate on the similarities or differences between their approach and these methods, particularly in the related works section?
The core strength of the paper is its strong theoretical foundation, presenting a novel idea of using Mahalanobis distance to extend the randomized smoothing framework, enabling it to handle complex adversarial attacks. It demonstrates originality by expanding randomized smoothing to incorporate a range of noise distributions beyond isotropic Gaussian noise, thereby providing robustness against multi-norm, multi-type adversarial threats, which traditional smoothing methods struggle to defend against. This is a valuable and important direction to explore in the context of randomized smoothing. The experimental results also support the theoretical claims, further adding to the strength of the paper.
This paper proposes *Dynamic Smoothing* (DSMOOTH), an extension of the randomized smoothing framework aimed at enhancing robustness against complex adversarial attacks. Traditional randomized smoothing methods rely on isotropic Gaussian noise, which limits their effectiveness against multi-norm and structured adversarial threats. Authors overcome these limitations by employing a broader range of noise distributions and using Mahalanobis distance to define probabilistic robustness guarantees, making it more adaptable to localized and non-uniform attacks. The authors validate DSMOOTH’s effectiveness through experiments on CIFAR-10 and IMAGENET, where it demonstrates significantly improved certified accuracy against state-of-the-art baselines in multi-attack scenarios.
While the paper presents promising theoretical advancements, several issues weaken its overall contribution and lead me to lean towards rejection. Although the experimental results support the theoretical claims, the setup itself lacks rigor. For instance, the evaluation is conducted on only 500 images from the CIFAR-10 test set rather than the full test set, potentially skewing the robustness results and limiting the generalizability of the findings. This choice is not well-justified, especially given the scale of CIFAR-10 and the availability of complete test sets. Additionally, the authors choose not to compare against baselines designed for $l_{p}$-norm defenses with $p > 2$, but the reasons for this omission are vague. This exclusion undermines the evaluation since these defenses are common benchmarks in adversarial robustness research. Further, while the paper aims to address robustness in a multi-attack setting, it only tests a single combination of attacks (Square Attack + FGSM). The limitation to just one multi-attack scenario significantly weakens the experimental results, as the approach’s effectiveness under diverse, real-world multi-attack combinations remains unverified. Expanding the experimental setup to include multiple combinations and comprehensive baseline comparisons would strengthen the paper’s contributions.

One more thing I would like to point out is that, the authors discuss isotropic gaussian distribution (this has been referred to as isotopic consistently in the paper which is wrong terminology), something similar has been discussed in few other works in the context of randomized smoothing [1,2], similaritities and/or dissimilarities with those methods is not discussed in the related works section. 

References:
[1] Hanbin Hong and Yuan Hong. Certified adversarial robustness via anisotropic randomized smoothing. arXiv preprint arXiv:2207.05327, 2022.
[2] Francisco Eiras, Motasem Alfarra, M Pawan Kumar, Philip HS Torr, Puneet K Dokania, Bernard Ghanem, and Adel Bibi. Ancer: Anisotropic certification via sample-wise volume maximization. arXiv preprint arXiv:2107.04570, 2021.
Yes
No ethics review needed.
1. Add reference to the work [1].
1. Clarify why the Mahalanobis-norm certified accuracy of DSMOOTH is compared to the L1 and L2-norm certified accuracy of other methods.
2. Discuss the rationale for using Mahalanobis Distance and identify practical scenarios where its robustness is applicable, assessing DSMOOTH's performance in those contexts. 

[1] ANCER: Anisotropic Certification via Sample-wise Volume Maximization. PMLR 2022
The paper introduces a approach to dynamic smoothing that adapts noise levels based on input complexity, distinguishing it from traditional static methods and offering a new direction in randomized smoothing.
The paper investigates the effectiveness of dynamic smoothing as a defense mechanism against complex adversarial attacks in machine learning models. The authors assert that existing defense strategies often fall short in providing certified robustness, particularly in the face of sophisticated adversarial inputs. Their key contributions include:
1.	Theoretical Analysis: The paper establishes a dynamic smoothing framework that adapts the noise level based on input complexity, proving that this approach enhances the model's certified robustness against a variety of adversarial attacks.
2.	Empirical Findings: The paper demonstrate through extensive experiments that the proposed dynamic smoothing method significantly improves robustness compared to traditional static smoothing techniques, while maintaining performance on both easy and hard samples.
3.	Proposed Solutions: The paper introduce specific training strategies, including adaptive noise levels and robust certification techniques, which allow for effective defense against complex attacks. These solutions not only enhance certified robustness but also improve overall model performance.
1. The scope of this paper is limited. Its primary contribution, anisotropic Gaussian randomized smoothing, has already been addressed by ANCER [1], which generalized Gaussian randomized smoothing to anisotropic Gaussian distributions and provided a robustness certification. This significantly diminishes the novelty of the current work. Moreover, the paper does not reference this closely related work [1].

2. The writing lacks clarity, as exemplified by the inclusion of the phrase "Code: [removed for review]" in the abstract; this should be omitted. The experimental setup is inadequately explained. The paper needs to clarify why the Mahalanobis-norm certified accuracy of DSMOOTH is compared to the L1 and L2-norm certified accuracy of other methods. It should also discuss the rationale for using Mahalanobis Distance and identify practical scenarios where its robustness is applicable, assessing DSMOOTH's performance in those contexts. 

3. The benefits introduced by the modified noise are not well-presented. Finally, unnecessary text in the summary should be removed.

[1] ANCER: Anisotropic Certification via Sample-wise Volume Maximization. PMLR 2022
Yes
No ethics review needed.
See " Experiments/Results Section" above under Weaknesses. There are several aspects of the experiments that could be clarified:
- It is currently unclear what quantity is being reported as "Certified Radius" for each defense method ( L1, L2, or Mahalanobis Distance radius). (It would be better to report "Certified Volumes" as in (Pfrommer et al 2023; Tecot 2021))
- What does "maximum perturbation 0.5" mean for the Square Attack?
- Are the attack perturbations $\delta$ used to compute $\Sigma$ based on attacks on the base classifier or the smoothed classifier?
- What models were used for the baseline smoothing techniques in the experimental section? Were these models also adversarially trained?
- How is  $\Sigma^{-1}$ computed with a low-rank  approximation of $\Sigma$?
The particular idea of smoothing using on a covariance matrix derived specifically from the empirical distribution of adversarial attack directions is, to my knowledge, novel, and may be promising.

The presentation of the theoretical results is clear for the most part.
This paper proposes a generalization of Randomized-Smoothing-based robustness certification. It first claims a very general result (Theorem 4.3), which states that if one applies randomized smothing (Cohen et al 2019) with the added Gaussian noise transformed by an arbitrary (invertible) function p,  then the resulting smoothed classifier will enjoy certified robustness under a metric determined by $p^{-1}.$ Precisely, for a smoothed classifier:

$$g(x) := \arg \max_{i \in [C]} P_{\epsilon \sim N(0,\sigma^2 I)} f(x + p(\epsilon))  = i), $$

if 

$$ P_{\epsilon\sim N(0,\sigma^2 I)} f(x + p(\epsilon))  =A )   \geq p_A \geq p_B \geq   \max_{B \neq A} P_{\epsilon \sim N(0,\sigma^2 I)} f(x + p(\epsilon))  =B)$$

Then, for any perturbation $\delta$ such that  $|p^{-1}(\delta)|_2 <  \sigma/2 * (\Phi^{-1}(p_A) - \Phi^{-1}(p_B))$, we have that $g(x+\delta) = A.$

This general result is then used to prove a more specific smoothing certificate (Lemma 4.5), claiming that if one applies randomized smoothing with anisotriopic Gaussian noise under an arbitrary covariance matrix $\Sigma$, then this results in certified robustness under the Mahalanobis metric defined by $\Sigma$:

$$\text{Mahalanobis}_\Sigma (\delta) := \sqrt{\delta^T \Sigma^{-1} \delta}$$

This more specific anisotropic smoothing result is proposed as the basis for a practical method,  D-Smooth, for certifiably robust classification. In D-Smooth, the smoothing covariance matrix is determined as the empirical covariance of adversarial attack perturbations computed from samples in the dataset. Note that because different attacks will result in different covariance matrices, this allows the smoothing distribution to be tailored to a specific attack or threat model. Experiments are conducted on CIFAR-10 and ImageNet, using an attack procedure that combines an L_2 and an L_infinity attack.
# Theoretical Claims

The more-general theoretical claim of this paper (Theorem 4.3) is incorrect, while the more specialized claim for anisotropic Gaussian smoothing (Lemma 4.5) can be proved directly with a much simpler proof than the one provided, and in fact appears in prior works.

- **Theorem 4.3 is incorrect**: one can construct a counterexample in one dimension. In particular, let:

$$
p(x) :=
\begin{cases}
x,&\text{if }x\geq 0\text{ or }x < -2\\\\
x -1,&\text{if } -1 \leq x < 0\\\\
x +1,&\text{if } -2 \leq x < -1
\end{cases}
$$
Note that $p$ is deterministic and invertible (in fact, $p^{-1} = p$), and let:

$$
f(x) :=
\begin{cases}
A,&\text{for }x\leq -1\text{ or }0 \leq x < 1\\\\
B,&\text{elsewhere.}\\\\
\end{cases}
$$

(Note, for later, that:
$$
f(p(\epsilon)) =
\begin{cases}
A,&\text{for }\epsilon\leq -2\text{ or }-1 \leq \epsilon < 1\\\\
B,&\text{elsewhere.}\\\\
\end{cases}
$$ and $$ f(p(\epsilon) + 0.5) =
\begin{cases}
A,&\text{for }\epsilon< -2\text{ or }-1.5 \leq \epsilon  \leq  0.5 \text{ or } 0 \leq \epsilon < 0.5 \\\\
B,&\text{elsewhere.}\\\\
\end{cases}
$$),
and let $\sigma = 1$.

Then, we consider the point $x = 0$ 
Then, 

 $p_A = P_{\epsilon\sim\mathcal{N}(0,1)} [f(x + p(\epsilon)) = A ] =  P_{\epsilon\sim\mathcal{N}(0,1)}  [f(p(\epsilon)) = A ]$

$ = \int^{-2}_{-\infty}  \text{NormPDF}(x) $ 

$+  \int^1_{-1}  \text{NormPDF}(x) \approx 0.7055$

 and $p_B = 1-p_A$

 Then, $\sigma/2 \cdot (\Phi^{-1}(p_A) - \Phi^{-1}(p_B)) = \Phi^{-1}(p_A) \approx 0.5403$, so Theorem 4.3 states that for any (1D) displacement vector $\delta$ such that: $|p^{-1}(\delta)|_2 = |p^{-1} (\delta)| \leq 0.54$, we should have that:
  $g(x + \delta)  = A$, 
which means that 

$ P_{\epsilon\sim\mathcal{N}(0,1)} [f(x + \delta+ p(\epsilon)) = A ]\geq 0.5$

However, if we plug back in $x = 0$ and use  displacement vector $\delta = 0.5$, so $|p^{-1} (\delta)| = |p^{-1} (0.5)| = 0.5 \leq 0.54$, we have:

 $p_A = P_{\epsilon\sim\mathcal{N}(0,1)} [f(x + \delta + p(\epsilon)) = A ] =  P_{\epsilon\sim\mathcal{N}(0,1)}  [f(p(\epsilon) + 0.5) = A ]$

$ = \int^{-2}_{-\infty}  \text{NormPDF}(x) $ 

$ + \int^{-0.5}_{-1.5}  \text{NormPDF}(x) $ 

$+  \int^{0.5}_{0}  \text{NormPDF}(x) \approx 0.4559 < 0.5$

Which is a contradiction.

- **The proof provided in this paper of Lemma 4.5 relies on  Theorem 4.3  and is therefore invalid. Despite this, Lemma 4.5, which is the result actually used for the proposed D-Smooth algorithm, happens to be true. However, it can be proven correct with just a few lines as a corollary of Cohen et al. 2019's theorem.**

In brief, starting with Cohen et al. 2019's theorem, with $\sigma = 1$:

for all $f : \mathbb{R}^d \rightarrow Y,\\, x \in  \mathbb{R}^d,$  and $\delta$ such that $|\delta|_2 < 1/2 \cdot (\Phi^{-1}(p_A) - \Phi^{-1}(p_B))$: 

$$P_{\epsilon\sim\mathcal{N}(0,1)} [f(x + \epsilon) = A ]   \geq p_A \geq p_B \geq \max_{B\neq A} P_{\epsilon\sim\mathcal{N}(0,1)} [f(x + \epsilon) = B ] $$ implies 

$$P_{\epsilon\sim\mathcal{N}(0,1)} [f(x + \delta+ \epsilon) = A ] > \max_{B \neq A}P_{\epsilon\sim\mathcal{N}(0,1)} [f(x + \delta+\epsilon) = B ]$$

If we fix the function $f$, and apply the theorem to the function $f'(x) := f(L(x))$, where $LL^T$ is the Cholesky decomposition of the matrix $\Sigma$, then we have, by linearity:

for all $f : \mathbb{R}^d \rightarrow Y,\\, x' \in  \mathbb{R}^d,$  and $\delta'$ such that $|\delta'|_2 < 1/2 \cdot (\Phi^{-1}(p_A) - \Phi^{-1}(p_B))$: 

$$P_{\epsilon\sim\mathcal{N}(0,1)} [f(Lx' + L\epsilon) = A ]     \geq p_A \geq p_B \geq \max_{B\neq A} P_{\epsilon\sim\mathcal{N}(0,1)} [f(Lx' + L\epsilon) = B ] $$ implies 

$$P_{\epsilon\sim\mathcal{N}(0,1)} [f(Lx' + L\delta'+ L\epsilon) = A ] > \max_{B \neq A}P_{\epsilon\sim\mathcal{N}(0,1)} [f(Lx' + L\delta'+L\epsilon) = B ]$$

Which can be written equivalently as 

for all $f : \mathbb{R}^d \rightarrow Y,\\, x' \in  \mathbb{R}^d,$  and $\delta'$ such that $|\delta'|_2 < 1/2 \cdot (\Phi^{-1}(p_A) - \Phi^{-1}(p_B))$: 

$$P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(Lx' + \epsilon) = A ]     \geq p_A \geq p_B \geq \max_{B\neq A} P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(Lx' + \epsilon) = B ] $$ implies 

$$P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(Lx' + L\delta'+ \epsilon) = A ] > \max_{B \neq A}P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(Lx' + L\delta'+\epsilon) = B ]$$

Then, for any x, we can take $x' :=  L^T \Sigma^{-1}  x$ , so that $Lx' = x$, and  we have:

For all $x \in  \mathbb{R}^d,$  and $\delta'$ such that $|\delta'|_2 < 1/2 \cdot (\Phi^{-1}(p_A) - \Phi^{-1}(p_B))$: 

$$P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(x + \epsilon) = A ]    \geq p_A \geq p_B \geq \max_{B\neq A} P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(x + \epsilon) = B ] $$ implies 

$$P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(x + L\delta'+ \epsilon) = A ] > \max_{B \neq A}P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(x + L\delta'+\epsilon) = B ]$$


Finally, for any $\delta$ such that $\text{Mahalanobis}_\Sigma (\delta) \leq  1/2 \cdot (\Phi^{-1}(p_A) - \Phi^{-1}(p_B))$,  we can take $\delta'  :=  L^T \Sigma^{-1}  \delta$ , so that $L \delta' = \delta$; this yields that  $|\delta'|_2 < 1/2 \cdot (\Phi^{-1}(p_A) - \Phi^{-1}(p_B))$. Then, we have:


For all $x \in  \mathbb{R}^d,$  and $\delta$ such that $\text{Mahalanobis}_\Sigma (\delta) \leq  1/2 \cdot (\Phi^{-1}(p_A) - \Phi^{-1}(p_B))$: 

$$P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(x + \epsilon) = A ]     \geq p_A \geq p_B \geq \max_{B\neq A} P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(x + \epsilon) = B ] $$ implies 

$$P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(x + \delta+ \epsilon) = A ] > \max_{B \neq A}P_{\epsilon\sim\mathcal{N}(0,\Sigma)} [f(x + \delta+\epsilon) = B ]$$
Scaling $\Sigma$ by scalar constants yields the lemma in the paper. It is important to notice that the linearity of the transformation was used here: this is why we can prove Lemma 4.5, even though the more general Theorem 4.3 is false.

(Closely related -- and even identical -- results appear in prior works which were not cited; see below). 

# Missing Related Work and Novelty

There are several works that have similar contributions to this work, but were not cited. The existance of these prior works reduces the novelty of the work; and relevant citations should be added.

As discussed above, Theorem 4.3 of this work is incorrect, so the first main correct contribution of this work is the smoothing certificate under anisotropic Gaussian noise (Lemma 4.5). A second contribution is the  proposed way of determining the covariance matrix Sigma to use for this certificate.

Considering first only the certificate itself (Lemma 4.5), the following similar works exist:

- Tecot 2021: Derives the same certificate in the coordinate-axis-aligned (diagonal covariance matrix) case. (However, this is a master's thesis, not a peer-reviewed paper, so it might not count as necessary prior work by ICLR standards)

- Pfrommer et al 2023: Derives a related certificate. Specifically, the proposed method projects all samples into a low-dimensional hyperplane before applying smoothing. This is equivalent to using a covariance matrix with eigenvalue 1 in some subset of directions, and eigenvalue "infinity" in all others.

- Eiras et al 2022: Corollary 1 of Eiras et al. is logically identical to Lemma 4.5 of the submission. However, in practice, Eiras et al only applies this result to the diagonal covariance matrix case. (Also, Eiras et al. proposes to use a covarance matrix that is _sample-dependent_, using a sceme introduced by Alfarra et al, 2022 for isotropic smoothing with sample-dependent scalar variance. Súkeník et al, 2022  [Appendix C] argues convincingly that the sample-dependent smoothing scheme proposed by Alfarra et al, 2022 is fundamentally unsound, and this flaw is inherited by Eiras et al 2022, so that their overall results are likely unsound. However, for a given matrix $\Sigma$, the fact remains that  Eiras et al. gives an equivalent lemma to to this work).

- Rumezhak et al 2023: followup work to  Eiras et al 2022, which uses the result for general covariance matrices (See Equations 2 and 3 of  Rumezhak et al 2023). However, this uses the same likely-flawed sample-dependence scheme as  Eiras et al 2022.

In terms of methods to compute the covariance matrix, none of the above works propose exactly the method used in this paper (computing the covariance of adversarial perturbations), so it may be novel. However, it should be compared empirically to the methods used in the above:

- Pfrommer et al 2023 uses the covariance of _clean_ samples in the dataset to determine the  subspace to project into.

- Tecot 2021 uses an optimization approach to determine a $\Sigma$ that maximizes the average certificate volume.

# Experiments/Results Section
The reported results show  _certified_ accuracies of a classifier smoothed with DSmooth on test samples form CIFAR-10 and ImageNet. This is compared to the certified accuracies of classifiers smoothed with prior techniques: Cohen et al. 2019's Gaussian randomized smoothing technique, which is certifiably robust under L2 perturbation, and Teng et al. 2020's Laplace Smoothing technique, which is certifiably robust under L1 perturbation.

However, I don't understand the experiment that is presented. In Figures 1 and 2, the x axis is just labeled as "certified radius". Is this the L1, L2, or Mahalanobis Distance radius? If it is any one of these (let's say, Mahalanobis Distance), how is the Mahalanobis certificate computed for, say, Cohen's method, which only provides L2 certificates? If the "certified radius" is in different metrics for each technique, then this isn't a fair apples-to-apples comparison. A method for comparing certificates under different metrics that has been used in prior work (ex. Pfrommer et al 2023; Tecot 2021) is to compare the _volumes_ of the certified regions: this would give a more fair comparison. Also, the paper mentions adversarially training the classifier. However, it's not clear if the baseline certificates were also for an adversairally trained model; if not, then the comparison is not fair. (See Salman et al 2019 for an adversairal training technique for isotropic randomized smoothing.)

Additionally, for the adversarial training / computation of $\Sigma$, it is stated on line 363-364: "In this experiment we use Square Attack with maximum perturbation 0.5 and 5000 queries." This is unclear. If the Square Attack is, as stated, an L_infinity-bounded attack, then "maximum perturbation 0.5" might mean one of two things: either (a) the distortion in each pixel is bounded by 0.5, where each pixel is normalized to [0,1]; or (b) the distortion in each pixel is bounded by 0.5, where each pixel is not normalized, and so lies in the discrete space {0,1,2,...,255}. If (a) is true, then this is a gigantic perturbation, and not an imperceptible adversarial attack.

Also, empirical accuracies under adversarial attack are not reported. In general, I do not think it is necessarily a requirement for certified robustness papers to report empirical accuracies of the classifiers under attack; however, it is important to make the claims made in the paper match the evidence. For instance, I think that the claim on lines 477-478: "this demonstrates that DSMOOTH is effective against complex adversarial attacks as defined in Def. 3.1, while RANDSMOOTH and LSMOOTH are inadequate for this purpose." is too strong, based on certification results alone. It is not tested how well any of these models _actually_ perform under attack.

Additionally, the comparison to Laplace noise (Teng et al. 2020) for the L1 metric is outdated: Yang et al 2020 has shown that uniform noise yeilds better L1 certificates, and Levine & Feizi 2021 propose an alternate "splitting noise" technique that also yeilds better certificates. Voracek and Hein (2023) make further improvements.

# Minor Issues
- ICLR call for papers allows unlimited appendices after references in the main PDF submission: I would encourage authors to move appendices here, instead of having them in a seperate file.
- Line 17; 'then' -> 'than'
- Cohen et al (2019) reference is duplicated: Cohen et al (2019a) and  Cohen et al (2019b) are the same
- Line 52: "defending against complex, high-dimensional adversarial attacks": Aren't almost all (norm-bounded) adversarial attacks high-dimensional? (In the sense that they have ther same dimensionality as the sample)
- It is generally considered standard to capitalize Equation in, for example, 'equation 1'
- The paragraph in lines 112-117 is not quite correct as written. The threat model as written in Equation 1  seems to only refer to _additive_ attacks: specifically, the perturbed sample is x + δ, where δ is constrained to be in the set C(δ). Note that as written, C(δ) does not seem to depend on on x, so some of the _non-additive_ threat models mentioned in the paragraph (for example, recoloring attacks and physical adeversarial attacks) are not encompased by this. This could be fixed by making  C(δ) depend on x, or (preferably) by replacing  x + δ with some more general 'perturbation function' q(x,δ).
- Line 125: "highly-dimensional" -> "high-dimensional"
- Line 136: I wouldn't use $\mathbb{P}$ to both mean Probability of an event, and to refer to a specific distribution.
- I'm confused about how the smoothing covariance Σ is constructed: it depends on the distribution of (optimal) attacks as given in Equation 1, which itself depends on the classifier being attacked. But Σ is part of the definition of the final smoothed classifier g, so this seems like it could be circular. Unless Σ is defined in terms of the attacks on the base classifier f? If so, this should be stated explicitly.
- Lines 191-197: If $\Sigma$ is not full-rank, won't $\Sigma$-inverse in definition of the Gaussian distribution (and Mahalanobis distance) not be defined? Doesn't this break the assumption in footnote 1?
 - Lines 391-392: " These are two NVIDIA GPUs with a 64-bit width and clock speed of 33 MHz, and four NVIDIA GPUs of the GV102 model with a 64-bit width, operating at a clock speed of 33 MHz." -- It seems that all 6 have a  clock speed of 33 MHz and 64-bit width. Why is the model number only mentioned for the first two? Also, I would double-check this for accuracy: 33 MHz is an extremely slow clock speed for a GPU, (from a brief search) all GV102 cards have base clock speeds far higher than this.

# References

Tecot, L. M. (2021). Robustness verification with non-uniform randomized smoothing. University of California, Los Angeles.

Pfrommer, S., Anderson, B., & Sojoudi, S. (2023). Projected Randomized Smoothing for Certified Adversarial Robustness. Transactions on Machine Learning Research.

Rumezhak, T., Eiras, F. G., Torr, P. H., & Bibi, A. (2023). RANCER: Non-axis aligned anisotropic certification with randomized smoothing. IEEE/CVF Winter Conference on Applications of Computer Vision.

Eiras, F., Alfarra, M., Torr, P., Kumar, M. P., Dokania, P. K., Ghanem, B., & Bibi, A. (2022). ANCER: Anisotropic Certification via Sample-wise Volume Maximization. Transactions on Machine Learning Research.

Alfarra, M., Bibi, A., Torr, P. H., & Ghanem, B. (2022). Data dependent randomized smoothing. Uncertainty in Artificial Intelligence.

Súkenı́k, P., Kuvshinov, A., & Günnemann, S. (2022, June). Intriguing Properties of Input-Dependent Randomized Smoothing. International Conference on Machine Learning.

Salman, H., Li, J., Razenshteyn, I., Zhang, P., Zhang, H., Bubeck, S., & Yang, G. (2019). Provably robust deep learning via adversarially trained smoothed classifiers. Advances in neural information processing systems

Yang, G., Duan, T., Hu, J. E., Salman, H., Razenshteyn, I., & Li, J. (2020). Randomized smoothing of all shapes and sizes. In International Conference on Machine Learning.

Levine, A. J., & Feizi, S. (2021). Improved, deterministic smoothing for l_1 certified robustness. International Conference on Machine Learning

Vorácek, V., & Hein, M. (2023). Improving l1-certified robustness via randomized smoothing by leveraging box constraints. International Conference on Machine Learning.
A novel approach based on radnomised smoothing to make vision classifiers robust against complex attacks.
@inproceedings{
anonymous2024certified,
title={Certified Defense Against Complex Adversarial Attacks with Dynamic Smoothing},
author={Anonymous},
booktitle={Submitted to The Thirteenth International Conference on Learning Representations},
year={2024},
url={https://openreview.net/forum?id=85Eej2kUHQ},
note={under review}
}
Randomized smoothing has emerged as a certified defence mechanism with probabilistic guarantees that works at scale. However, current randomized smoothing methods offer theoretical guarantees that are limited by their reliance on specific noise distributions, and they struggle to handle complex adversarial attacks. In this paper, we propose a novel certification method based on randomized smoothing designed to handle complex adversarial attacks, including combinations of multiple attack types. We call this method Dynamic Smoothing (DSmooth). Our key idea is to incorporate more general distributions for smoothing then isotopic Gaussian noise, for which probabilistic guarantees can be derived in terms of the Mahalanobis distance. These general distributions make the smoothed classifier more robust against a wide range of threats, including localized adversarial attacks and multi-attacks. We validate the performance of our method experimentally on challenging threat models using CIFAR-10 and ImageNet, and demonstrate its superiority over state-of-the-art defenses in terms of certified accuracy. Our results show that the proposed method significantly improves the robustness of machine learning models against complex attacks, advancing their suitability for use in safety-critical applications. Code: [removed for review]
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Certified Defense Against Complex Adversarial Attacks with Dynamic Smoothing
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