Robustness of Classifiers to Universal Perturbations: A Geometric Perspective
Deep networks have recently been shown to be vulnerable to universal perturbations: there exist very small image-agnostic perturbations that cause most natural images to be misclassified by such classifiers. In this paper, we provide a quantitative analysis of the robustness of classifiers to universal perturbations, and draw a formal link between the robustness to universal perturbations, and the geometry of the decision boundary. Specifically, we establish theoretical bounds on the robustness of classifiers under two decision boundary models (flat and curved models). We show in particular that the robustness of deep networks to universal perturbations is driven by a key property of their curvature: there exist shared directions along which the decision boundary of deep networks is systematically positively curved. Under such conditions, we prove the existence of small universal perturbations. Our analysis further provides a novel geometric method for computing universal perturbations, in addition to explaining their properties.
 The paper is written well and clear. The core contribution of the paper is the illustration that: under the assumption of flat, or curved decision boundaries with positive curvature small universal adversarial perturbations exist. Pros: the intuition and geometry is rather clearly presented. Cons: References to "CaffeNet" and "LeNet" (even though the latter is well-known) are missing. In the experimental section used to validate the main hypothesis that the deep networks have positive curvature decision boundaries, there is no description of how these networks were trained. It is not clear why the authors have decided to use out-dated 5-layer "LeNet" and NiN (Network in network) architectures instead of more recent and much better performing architectures (and less complex than NiN architectures). It would be nice to see how the behavior and boundaries look in these cases. The conclusion is speculative: "Our analysis hence shows that to construct classifiers that are robust to universal perturbations, it is key to suppress this subspace of shared positive directions, which can possibly be done through regularization of the objective function. This will be the subject of future works." It is clear that regularization should play a significant role in shaping the decision boundaries. Unfortunately, the paper does not provide details at the basic level, which algorithms, architectures, hyper-parameters or regularization terms are used. All these factors should play a very significant role in the experimental validation of their hypothesis. Notes: I did not check the proofs of the theorems in detail.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Robustness of Classifiers to Universal Perturbations: A Geometric Perspective
Deep networks have recently been shown to be vulnerable to universal perturbations: there exist very small image-agnostic perturbations that cause most natural images to be misclassified by such classifiers. In this paper, we provide a quantitative analysis of the robustness of classifiers to universal perturbations, and draw a formal link between the robustness to universal perturbations, and the geometry of the decision boundary. Specifically, we establish theoretical bounds on the robustness of classifiers under two decision boundary models (flat and curved models). We show in particular that the robustness of deep networks to universal perturbations is driven by a key property of their curvature: there exist shared directions along which the decision boundary of deep networks is systematically positively curved. Under such conditions, we prove the existence of small universal perturbations. Our analysis further provides a novel geometric method for computing universal perturbations, in addition to explaining their properties.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



 The paper is written well and clear. The core contribution of the paper is the illustration that: under the assumption of flat, or curved decision boundaries with positive curvature small universal adversarial perturbations exist. Pros: the intuition and geometry is rather clearly presented. Cons: References to "CaffeNet" and "LeNet" (even though the latter is well-known) are missing. In the experimental section used to validate the main hypothesis that the deep networks have positive curvature decision boundaries, there is no description of how these networks were trained. It is not clear why the authors have decided to use out-dated 5-layer "LeNet" and NiN (Network in network) architectures instead of more recent and much better performing architectures (and less complex than NiN architectures). It would be nice to see how the behavior and boundaries look in these cases. The conclusion is speculative: "Our analysis hence shows that to construct classifiers that are robust to universal perturbations, it is key to suppress this subspace of shared positive directions, which can possibly be done through regularization of the objective function. This will be the subject of future works." It is clear that regularization should play a significant role in shaping the decision boundaries. Unfortunately, the paper does not provide details at the basic level, which algorithms, architectures, hyper-parameters or regularization terms are used. All these factors should play a very significant role in the experimental validation of their hypothesis. Notes: I did not check the proofs of the theorems in detail.
assistant
 The paper is written well and clear. The core contribution of the paper is the illustration that: under the assumption of flat, or curved decision boundaries with positive curvature small universal adversarial perturbations exist. Pros: the intuition and geometry is rather clearly presented. Cons: References to "CaffeNet" and "LeNet" (even though the latter is well-known) are missing. In the experimental section used to validate the main hypothesis that the deep networks have positive curvature decision boundaries, there is no description of how these networks were trained. It is not clear why the authors have decided to use out-dated 5-layer "LeNet" and NiN (Network in network) architectures instead of more recent and much better performing architectures (and less complex than NiN architectures). It would be nice to see how the behavior and boundaries look in these cases. The conclusion is speculative: "Our analysis hence shows that to construct classifiers that are robust to universal perturbations, it is key to suppress this subspace of shared positive directions, which can possibly be done through regularization of the objective function. This will be the subject of future works." It is clear that regularization should play a significant role in shaping the decision boundaries. Unfortunately, the paper does not provide details at the basic level, which algorithms, architectures, hyper-parameters or regularization terms are used. All these factors should play a very significant role in the experimental validation of their hypothesis. Notes: I did not check the proofs of the theorems in detail.