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Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning

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  • Bolte, Jérôme
  • Pauwels, Edouard

Abstract

Modern problems in AI or in numerical analysis require nonsmooth approaches with a exible calculus. We introduce generalized derivatives called conservative fields for which we develop a calculus and provide representation formulas. Functions having a conservative field are called path differentiable: convex, concave, Clarke regular and any semialgebraic Lipschitz continuous functions are path differentiable. Using Whitney stratification techniques for semialgebraic and definable sets, our model provides variational formulas for nonsmooth automatic diffrentiation oracles, as for instance the famous backpropagation algorithm in deep learning. Our differential model is applied to establish the convergence in values of nonsmooth stochastic gradient methods as they are implemented in practice.

Suggested Citation

  • Bolte, Jérôme & Pauwels, Edouard, 2019. "Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning," TSE Working Papers 19-1044, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:123631
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    References listed on IDEAS

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    1. Bolte, Jérôme & Castera, Camille & Pauwels, Edouard & Févotte, Cédric, 2019. "An Inertial Newton Algorithm for Deep Learning," TSE Working Papers 19-1043, Toulouse School of Economics (TSE).
    2. Michel Benaim & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions II: Applications," Levine's Bibliography 784828000000000098, UCLA Department of Economics.
    3. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions; Part II: Applications," Working Papers hal-00242974, HAL.
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    Cited by:

    1. Bolte, Jérôme & Pauwels, Edouard, 2021. "A mathematical model for automatic differentiation in machine learning," TSE Working Papers 21-1184, Toulouse School of Economics (TSE).
    2. Le, Tam & Bolte, Jérôme & Pauwels, Edouard, 2022. "Subgradient sampling for nonsmooth nonconvex minimization," TSE Working Papers 22-1310, Toulouse School of Economics (TSE).

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