Universal Approximation with Certified Networks
Training neural networks to be certifiably robust is critical to ensure their safety against adversarial attacks. However, while promising, state-of-the-art results with certified training are far from satisfactory. Currently, it is very difficult to train a neural network that is both accurate and certifiably robust. In this work we take a step towards addressing this challenge. We prove that for every continuous function , there exists a network such that: (i) approximates arbitrarily close, and (ii) simple interval bound propagation of a region through yields a result that is arbitrarily close to the optimal output of on . Our result can be seen as a Universal Approximation Theorem for interval-certified ReLU networks. To the best of our knowledge, this is the first work to prove the existence of accurate, interval-certified networks.
This paper proves the universal approximation property of interval-certified ReLU networks. Such a network achieves the best certified accuracy against l_infty adversaries at large epsilons on MNIST and CIFAR10, but the certified accuracy is still far from satisfaction. With the results from this paper, we can at least make sure the interval-certified ReLU networks is able to fit the robust classifier with arbitrarily small error if it exists. I did not check the details of the proofs, but I think such a theorem is a great reference for future work. However, we cannot conclude whether a network with a moderate size exists to fit the robust classifier well from this paper. From the construction of local bumps (nmin), the number of neurons needed to represent a bump function grows exponentially with the input dimension, which makes the size of the network vacuously large when the input is images. We are not getting any closer to improving the certified robustness with this paper.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Universal Approximation with Certified Networks
Training neural networks to be certifiably robust is critical to ensure their safety against adversarial attacks. However, while promising, state-of-the-art results with certified training are far from satisfactory. Currently, it is very difficult to train a neural network that is both accurate and certifiably robust. In this work we take a step towards addressing this challenge. We prove that for every continuous function , there exists a network such that: (i) approximates arbitrarily close, and (ii) simple interval bound propagation of a region through yields a result that is arbitrarily close to the optimal output of on . Our result can be seen as a Universal Approximation Theorem for interval-certified ReLU networks. To the best of our knowledge, this is the first work to prove the existence of accurate, interval-certified networks.
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>



This paper proves the universal approximation property of interval-certified ReLU networks. Such a network achieves the best certified accuracy against l_infty adversaries at large epsilons on MNIST and CIFAR10, but the certified accuracy is still far from satisfaction. With the results from this paper, we can at least make sure the interval-certified ReLU networks is able to fit the robust classifier with arbitrarily small error if it exists. I did not check the details of the proofs, but I think such a theorem is a great reference for future work. However, we cannot conclude whether a network with a moderate size exists to fit the robust classifier well from this paper. From the construction of local bumps (nmin), the number of neurons needed to represent a bump function grows exponentially with the input dimension, which makes the size of the network vacuously large when the input is images. We are not getting any closer to improving the certified robustness with this paper.
assistant
This paper proves the universal approximation property of interval-certified ReLU networks. Such a network achieves the best certified accuracy against l_infty adversaries at large epsilons on MNIST and CIFAR10, but the certified accuracy is still far from satisfaction. With the results from this paper, we can at least make sure the interval-certified ReLU networks is able to fit the robust classifier with arbitrarily small error if it exists. I did not check the details of the proofs, but I think such a theorem is a great reference for future work. However, we cannot conclude whether a network with a moderate size exists to fit the robust classifier well from this paper. From the construction of local bumps (nmin), the number of neurons needed to represent a bump function grows exponentially with the input dimension, which makes the size of the network vacuously large when the input is images. We are not getting any closer to improving the certified robustness with this paper.