Upper and lower bounds for the Lipschitz constant of random neural networks
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909672524480512 |
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| author | Geuchen, Paul Stöger, Dominik Telaar, Thomas Voigtlaender, Felix |
| author_facet | Geuchen, Paul Stöger, Dominik Telaar, Thomas Voigtlaender, Felix |
| contents | Empirical studies have widely demonstrated that neural networks are highly sensitive to small, adversarial perturbations of the input. The worst-case robustness against these so-called adversarial examples can be quantified by the Lipschitz constant of the neural network. In this paper, we study upper and lower bounds for the Lipschitz constant of random ReLU neural networks. Specifically, we assume that the weights and biases follow a generalization of the He initialization, where general symmetric distributions for the biases are permitted. For deep networks of fixed depth and sufficiently large width, our established upper bound is larger than the lower bound by a factor that is logarithmic in the width. In contrast, for shallow neural networks we characterize the Lipschitz constant up to an absolute numerical constant that is independent of all parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01356 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Upper and lower bounds for the Lipschitz constant of random neural networks Geuchen, Paul Stöger, Dominik Telaar, Thomas Voigtlaender, Felix Machine Learning Probability 68T07, 26A16, 60B20, 60G15 Empirical studies have widely demonstrated that neural networks are highly sensitive to small, adversarial perturbations of the input. The worst-case robustness against these so-called adversarial examples can be quantified by the Lipschitz constant of the neural network. In this paper, we study upper and lower bounds for the Lipschitz constant of random ReLU neural networks. Specifically, we assume that the weights and biases follow a generalization of the He initialization, where general symmetric distributions for the biases are permitted. For deep networks of fixed depth and sufficiently large width, our established upper bound is larger than the lower bound by a factor that is logarithmic in the width. In contrast, for shallow neural networks we characterize the Lipschitz constant up to an absolute numerical constant that is independent of all parameters. |
| title | Upper and lower bounds for the Lipschitz constant of random neural networks |
| topic | Machine Learning Probability 68T07, 26A16, 60B20, 60G15 |
| url | https://arxiv.org/abs/2311.01356 |