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Local Lipschitz Bounds of Deep Neural Networks

Author

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  • Calypso Herrera
  • Florian Krach
  • Josef Teichmann

Abstract

The Lipschitz constant is an important quantity that arises in analysing the convergence of gradient-based optimization methods. It is generally unclear how to estimate the Lipschitz constant of a complex model. Thus, this paper studies an important problem that may be useful to the broader area of non-convex optimization. The main result provides a local upper bound on the Lipschitz constants of a multi-layer feed-forward neural network and its gradient. Moreover, lower bounds are established as well, which are used to show that it is impossible to derive global upper bounds for the Lipschitz constants. In contrast to previous works, we compute the Lipschitz constants with respect to the network parameters and not with respect to the inputs. These constants are needed for the theoretical description of many step size schedulers of gradient based optimization schemes and their convergence analysis. The idea is both simple and effective. The results are extended to a generalization of neural networks, continuously deep neural networks, which are described by controlled ODEs.

Suggested Citation

  • Calypso Herrera & Florian Krach & Josef Teichmann, 2020. "Local Lipschitz Bounds of Deep Neural Networks," Papers 2004.13135, arXiv.org, revised Feb 2023.
  • Handle: RePEc:arx:papers:2004.13135
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    File URL: http://arxiv.org/pdf/2004.13135
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    References listed on IDEAS

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    1. Christa Cuchiero & Wahid Khosrawi & Josef Teichmann, 2020. "A Generative Adversarial Network Approach to Calibration of Local Stochastic Volatility Models," Risks, MDPI, vol. 8(4), pages 1-31, September.
    2. Christa Cuchiero & Wahid Khosrawi & Josef Teichmann, 2020. "A generative adversarial network approach to calibration of local stochastic volatility models," Papers 2005.02505, arXiv.org, revised Sep 2020.
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    Cited by:

    1. Calypso Herrera & Florian Krach & Pierre Ruyssen & Josef Teichmann, 2021. "Optimal Stopping via Randomized Neural Networks," Papers 2104.13669, arXiv.org, revised Dec 2023.
    2. Calypso Herrera & Florian Krach & Josef Teichmann, 2020. "Neural Jump Ordinary Differential Equations: Consistent Continuous-Time Prediction and Filtering," Papers 2006.04727, arXiv.org, revised Apr 2021.
    3. Calypso Herrera & Florian Krach & Anastasis Kratsios & Pierre Ruyssen & Josef Teichmann, 2020. "Denise: Deep Robust Principal Component Analysis for Positive Semidefinite Matrices," Papers 2004.13612, arXiv.org, revised Jun 2023.

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