Lipschitz constant estimation for Neural Networks via sparse polynomial optimization
We introduce LiPopt, a polynomial optimization framework for computing increasingly tighter upper bound on the Lipschitz constant of neural networks. The underlying optimization problems boil down to either linear (LP) or semidefinite (SDP) programming. We show how to use structural properties of the network, such as sparsity, to significantly reduce the complexity of computation. This is specially useful for convolutional as well as pruned neural networks. We conduct experiments on networks with random weights as well as networks trained on MNIST, showing that in the particular case of the -Lipschitz constant, our approach yields superior estimates as compared to other baselines available in the literature.
The authors study the problem of estimating the Lipschitz constant of a deep neural network with ELO activation function. The authors formulate the problem as a polynomial optimisation problem, which is elegant. Subsequently, they utilise an LP hierarchy based on the Krivine-Vasilescu-Handelman’s Positivstellensatz and suggest exploiting sparsity therein. The computational results are clearly not sufficient to apply this approach to real-world neural networks, but are still respectable. Section 3 (Theorem 2) is not original work, as leaving the theorem without a reference would imply: the authors cite Section 9 of Lasserre*s 2015 book later, so they are clearly aware of this, and there are many application even within verification, e.g., https://link.springer.com/content/pdf/10.1007%2F978-3-319-48989-6_44.pdf https://ieeexplore.ieee.org/document/8493559 The suggestions as to the exploitation of sparsity (Section 4) are not original work either. The authors could cite, e.g., JB Lasserre: Convergent SDP-relaxations in polynomial optimization with sparsity (SIAM Journal on Optimization, 2006), as one of the early proponents of the exploitation of sparsity. In Section 7: -- The claim *We observed clear improvement of the Lipschitz bound obtained, compared to the SDP method* is not supported by the results the authors present. -- The authors do not present the run-time. This needs to be included, considering they imply that the key improvement over the traditional SDP is that this works with smaller variables and should be faster. -- The presentation of the experimental results should be improved, so as to follow the NIPS reproducibility checklist, or at least have error bars at one standard deviation and standard deviation in the table. Other than that, the paper is well written (modulo Section missing in *Section 5* at the top of Section 7), and I would recommend its acceptance.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Lipschitz constant estimation for Neural Networks via sparse polynomial optimization
We introduce LiPopt, a polynomial optimization framework for computing increasingly tighter upper bound on the Lipschitz constant of neural networks. The underlying optimization problems boil down to either linear (LP) or semidefinite (SDP) programming. We show how to use structural properties of the network, such as sparsity, to significantly reduce the complexity of computation. This is specially useful for convolutional as well as pruned neural networks. We conduct experiments on networks with random weights as well as networks trained on MNIST, showing that in the particular case of the -Lipschitz constant, our approach yields superior estimates as compared to other baselines available in the literature.
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<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 authors study the problem of estimating the Lipschitz constant of a deep neural network with ELO activation function. The authors formulate the problem as a polynomial optimisation problem, which is elegant. Subsequently, they utilise an LP hierarchy based on the Krivine-Vasilescu-Handelman’s Positivstellensatz and suggest exploiting sparsity therein. The computational results are clearly not sufficient to apply this approach to real-world neural networks, but are still respectable. Section 3 (Theorem 2) is not original work, as leaving the theorem without a reference would imply: the authors cite Section 9 of Lasserre*s 2015 book later, so they are clearly aware of this, and there are many application even within verification, e.g., https://link.springer.com/content/pdf/10.1007%2F978-3-319-48989-6_44.pdf https://ieeexplore.ieee.org/document/8493559 The suggestions as to the exploitation of sparsity (Section 4) are not original work either. The authors could cite, e.g., JB Lasserre: Convergent SDP-relaxations in polynomial optimization with sparsity (SIAM Journal on Optimization, 2006), as one of the early proponents of the exploitation of sparsity. In Section 7: -- The claim *We observed clear improvement of the Lipschitz bound obtained, compared to the SDP method* is not supported by the results the authors present. -- The authors do not present the run-time. This needs to be included, considering they imply that the key improvement over the traditional SDP is that this works with smaller variables and should be faster. -- The presentation of the experimental results should be improved, so as to follow the NIPS reproducibility checklist, or at least have error bars at one standard deviation and standard deviation in the table. Other than that, the paper is well written (modulo Section missing in *Section 5* at the top of Section 7), and I would recommend its acceptance.
assistant
The authors study the problem of estimating the Lipschitz constant of a deep neural network with ELO activation function. The authors formulate the problem as a polynomial optimisation problem, which is elegant. Subsequently, they utilise an LP hierarchy based on the Krivine-Vasilescu-Handelman’s Positivstellensatz and suggest exploiting sparsity therein. The computational results are clearly not sufficient to apply this approach to real-world neural networks, but are still respectable. Section 3 (Theorem 2) is not original work, as leaving the theorem without a reference would imply: the authors cite Section 9 of Lasserre*s 2015 book later, so they are clearly aware of this, and there are many application even within verification, e.g., https://link.springer.com/content/pdf/10.1007%2F978-3-319-48989-6_44.pdf https://ieeexplore.ieee.org/document/8493559 The suggestions as to the exploitation of sparsity (Section 4) are not original work either. The authors could cite, e.g., JB Lasserre: Convergent SDP-relaxations in polynomial optimization with sparsity (SIAM Journal on Optimization, 2006), as one of the early proponents of the exploitation of sparsity. In Section 7: -- The claim *We observed clear improvement of the Lipschitz bound obtained, compared to the SDP method* is not supported by the results the authors present. -- The authors do not present the run-time. This needs to be included, considering they imply that the key improvement over the traditional SDP is that this works with smaller variables and should be faster. -- The presentation of the experimental results should be improved, so as to follow the NIPS reproducibility checklist, or at least have error bars at one standard deviation and standard deviation in the table. Other than that, the paper is well written (modulo Section missing in *Section 5* at the top of Section 7), and I would recommend its acceptance.