Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning
Modern problems in AI or in numerical analysis require nonsmooth approaches with a flexible 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 differentia...
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| Veröffentlicht in: | Mathematical programming Jg. 188; H. 1; S. 19 - 51 |
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01.07.2021
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| Abstract | Modern problems in AI or in numerical analysis require nonsmooth approaches with a flexible 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 differentiation 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. |
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| AbstractList | Modern problems in AI or in numerical analysis require nonsmooth approaches with a flexible 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 differentiation 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. |
| Author | Bolte, Jérôme Pauwels, Edouard |
| Author_xml | – sequence: 1 givenname: Jérôme surname: Bolte fullname: Bolte, Jérôme email: jerome.bolte@tse-fr.eu organization: Toulouse School of Economics, Université Toulouse 1 Capitole – sequence: 2 givenname: Edouard surname: Pauwels fullname: Pauwels, Edouard organization: IRIT, Université de Toulouse, CNRS, DEEL IRT Saint Exupery |
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| Keywords | o-Minimal structures 62M45 Neural nets and related approaches to inference from stochastic processes 90C06 Large-scale problems 68T05 Learning and adaptive systems in artificial intelligence Stochastic gradient Automatic differentiation Deep learning Backpropagation algorithm Clarke subdifferential 49M27 Decomposition methods First order methods Nonsmooth stochastic optimization Definable sets 65K10 Numerical optimization and variational techniques 49J53 Set-valued and variational analysis |
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| SubjectTerms | Algorithms Back propagation Calculus Calculus of Variations and Optimal Control; Optimization Combinatorics Continuity (mathematics) Deep learning Differentiation Full Length Paper Machine learning Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Theoretical |
| Title | Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning |
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